- Email: [email protected]
Smooth invariant curves of singularities of vector fields on ℝ3
Ann. lnsl. Henri
Vol.
3,
n°
Poincare,
2, 1986,
p. 111-183.
mooth invariant
curves
of
singularities
of vector fields
on
R3
Patrick BONCKAERT Limburgs
Universitair
B-3610
centrum,
universnaire
campus,
Diepenbeek (Belgium)
We describe singularities of Coo vector fields, mainly ABSTRACT. in ~3, in the neighbourhood of a given « generalized » direction (this is : the image of a Coo germ y : ([R + , 0) - (~3, 0) with y’(o) ~ 0). It is a local study. One of the major results is : if X is a germ in 0 E f~ 3 of a C~° vector field and if X is not infinitely flat along a direction D then there exists a cone of finite contact around D in which four specific situations can occur. In three of these situations D is formally invariant under X (with formally we mean : up to the level of formal Taylor series) and there exists a Coo one-dimensional invariant manifold having infinite contact with D. In particular, we obtain that the existence of a formally invariant direction D always implies the existence of a « real life » invariant direction having infinite contact with D, provided that X is not infinitely flat along D. Using the blowing up method for singularities of vector fields we reduce a singularity always to either a « flow box » or to a singularity with nonzero I-jet and with a formally invariant direction. Finally we give topological models for the obtained situations. I would like to thank Freddy Dumortier for suggesting me the problem and for his valuable help. -
RESUME. dans [R3 au
Nous decrivons les singularites de champs de vecteurs Cx voisinage d’une « direction » donnee, c’est-a-dire de l’image
-
Liste de mots-cles : Vector Field.
Singularity. Invariant Manifold. Classification AMS : 34C05 Ordinary Differential Equations. Qualitative Theory. Location of integral curves, singular points, limit cycles. Analyse non linéaire 86/02.111 / 73 S 9,301 ç Gauthier-Villars
Annales de l’Institut Henri Poincaré -
Vol.
3. 0294-1449
2
?. BONCKAERT
(R3, 0),
avec 03B3’(0) ~ 0. C’est une étude locale. si particulier que X est le germe a l’origine d’un champ C" ;i X n’est pas plat dans une direction D, alors il existe un cone de contact ~rdre fini autour de D ou quatre situations distinctes peuvent se produire. ms trois d’entre elles, D est formellement invariant par X et il existe e courbe invariante C~ presentant un contact d’ordre infini avec D. utilisant la methode de blow-up pour les singularites de champs de ~teurs, nous reduisons toutes les singularites soit a un « flow box », t a une singularite au 1-jet non trivial possedant une direction forllement invariante. Enfin, nous fournissons des modeles topologiques ur les diverses situations obtenues.
03B3 : (R+, 0)
m
germe
>us
montrons en
-
,
Je remercie Freddy Dumortier pour m’avoir ur son aide.
I.
suggere
ce
probleme
et
INTRODUCTION, PRELIMINARIES AND STATEMENT OF THE MAIN RESULT
§ 1. Elementary definitions
and useful theorems.
Let X be a vector field on ~n. We (1.1) DEFINITION. s a singularity in p E f~n if X(p) 0. If we only investigate local properties of a singularity in p, ction to put p in 0, the origin. -
say that X
=
it is
no res-
DEFINITION. Two vector fields X and Y on tR" are called germuivalent in 0 if there is a neighbourhood U of 0 on which they coincide,
( 1. 2)
-
DEFINITION. The set of all vector fields on [?" which are germuivalent with X (in 0) is called the germ of X (in 0). In the same way we n define germs in 0 of functions, diffeomorphism,... The germ in 0 of a set A ci {?" is the germ in 0 of its characteristic func>n A germ will often be confused with a representative of it if this
( 1. 3)
-
without
danger.
NOTATION. lds X on ~n with
( 1. 4)
-
Gn denotes the set of all germs in 0 of ex vector 0.
X(0)
=
DEFINITION. Let X, uivalent if their derivatives in 0 up to, and X(0) D’Y(0), Vi, 0 i k.
( 1. 5)
-
X and Y are called k-jet order k are equal:
including,
=
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
!R 3
113
The set of all Y E Gn which are k-jet equivalent It is important to with X E G" is called the k-jet of X and denoted observe that the k-th order Taylor approximation of X in 0 belongs to we often will make no distinction between these two objects. In 1 correspondence between k-jets and vector fields X fact there is a 1 0 and with polynomial component functions of degree k. on [Rn with X(0)
(1.6)
DEFINITION.
-
-
=
(1.7) NOTATION limit of the sets Ji for the
We
mappings nlk : J?
can
take the inverse -
-
jkX(0) (1
>
k).
DEFINITION. This inverse limit is denoted J; its elements called oo-jets; the oo-jet corresponding to is denoted Elements of J~ can be regarded as n-tupels of formal power series in n variables; also j~X(0) represents the Taylor series of X at 0. In the same way we define jets of functions, diffeomorphisms...
(1.8)
-
are
(1.9) and
DEFINITION. 1 X(0) ~ 0.
X E Gn is said to
-
haveflatness
k if
=
0
Y are said to be Cr conjuif for some (and hence for all) represen1) tatives X and Y of X resp. Y there are open neighbourhoods U and V of 0 in Rn and a C7 diffeomorphism 03C6 : U ~ V such that Vx E U :
( 1.10) DEFINITION. gate (r E N u { 00, cv },
-
X(x)
=
Let
X, Y E Gn. X and
r >
ifi( 03C6(x)).
We also write:
~~X
=
Y in this
case.
Let X, Y E Gn. X and Y are said to be Cr equi(1.11) DEFINITION. valent (r E N u { ~, 03C9}) if for some (and hence for all) representatives X -
and Y of X resp. Y there are open neighbourhoods U and V of 0 in fR" and a Cr diffeomorphism h : U ~ V which maps integral curves of X to integral curves of Y preserving the « sense » but not necessarily the para[0, t ]) c U, t > 0, then metrization ; more precisely : if p E U and and q,v there is some t’ > 0 such that [0, t’ ]) h( [0, t ])). denote the flow of X resp. Y). Time preserving Cr-equivalence is called Cr conjugacy and coincides with definition 1.10 for r > 1. With C° diffeomorphism we mean a homeomorphism. =
(1.12) THEOREM (Borel) [Di, Na]. - For all such that T = j~X(0). (Or: j x : Gn ~ J) is
T e J) there exists
surjective.).
an
X E Gn
0
in ofor shortlya direction, (1point.C’-difieomorphic 13) DEFINITION. image A generalized direction diffeomorphism is
a
0. The
Vol. 3. n° 2-1986.
of the germ in 0 E of a halfline with endis assumed to preserve 0.
14
P. BONCKAERT
( 1.14) DEFINITION. - A germ in 0 of a set K m [Rm+ 1 is called a C‘ l > k, if there exists a germ of a Cj function h : ~olid) cone of contact k, oo with (~0) jkh(o) 0, ~+1~(0) ~ 0 and a germ of a C" [0, [, 0) (Rm+ 1, 0) such that iffeomorphism 03C6: (!Rm + 1, 0) -
=
-
x K is called a cone of contact k around the direction D 4>( { The intersection of a C7 diffeomorphic image of a hyperplane, through D, with K is called a Crm-dimensional subcone of K (0
[0, 00 [) passing r 1).
=
If D is a direction in [?", then there exists a germ of a map y : ([0, oo [, 0) - (~", 0) with y’(0) # 0 such that the image of y is D. Conversely, if y : ( [0, oo [, 0) - (~ 0) is Coo and if y’(0) # 0, then the image of y is a direction.
(1.15)
PROPERTY.
-
Proof 2014 Easy.
D
(1. 16) DEFINITION. - For XEGn and D a direction we say that X is nonflat along D if for some (and hence for all) Coo germ }’ : ( [0, CfJ [0) with y’(0) # 0 and image D we have. y)(0) ~ 0. In the other case we say that X isflat along D. ~
We say that X E Gn satisfies a ojasiewicz (1.17) DEFINITION. quality if there exist k~N and C, 03B4 E ]0, 00 [ such that -
PROPERTY. - If X E Gn satisfies non-flat along every direction.
(1.18) s
( 1.19) CONVENTION. or
a
-
vector field X on ~m
( 1. 20) DEFINITION.
formally
-
a
Zojasiewicz inequality
then X
We write (x, z) for an element of [Rm x; we write X (Xx, Xz). =
We say that X E Gm + 1 leaves the
invariant if for all
ine-
(0)
=
z-axis {0 }m x R
O. This is the
same as
saying
does not contain pure z-terms. 1 x [0, oo D is formally invariant under X E Gm+ A direction D ~( { ~ * 1 X leaves the z-axis formally invariant. =
f
( 1. 21 ) THEOREM (Normal Form Theorem, Poincare and Dulac). - Let (E Gn and let Xi be the linear vector field on [?" such H~ denote the vector space of those vector fields on !?" Let, for vhose coefficient functions are homogeneous polynomials of degree h. Denote [X 1, - ]h : H - Hh : Y --~ [X 1, Y ] and let Bh - Im ( [X 1, - ]h ). Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
Choose for each h > 2
an
[R~
115
arbitrary supplementary space Gh for Bh in H~
(that is: Hh - Bh 0 Gh). Then there exists a germ of a C~ diffeomorphism such that ~,~X has an oo jet of the form
where gh E
h
Application (to
=
2, 3,
0)
-
....
be used later
G3 with Let X E 03
03C6:(Rn,
j1X(0)) (
=
on).
ax~ ~x,
a
O. If the z-axis 0.
012 0}2 R is
is
x
invariant under X, then there exists a diffeomorphism r~ : ([R3, 0) X’ has an ~ jet of the form such that
formally -
([R3, o)
=
,
Proof [Ta ]. Suppose, by induction, that for i E N, i X = X+
g2 +
...
write Rl Y E Hi with [Xi, Y]
We
can
>
1,
one can
write
with the gh~Gh and with ji-1Ri-1(0)=0. gi E Gi, jiRi(o) 0. Take an
+ gi-1 + R1-1 -1 - gi + b; with
=
bi. Taking §;= ~Y(., 2014 1 ) (the time -1 mapping of Y) one can check that + gi + Ri: (4)d*(X) == X1 + g2 + Since ji-1 ji-1 Id (0) we can apply the theorem of Borel (for diffeomorphisms) to obtain the desired diffeomorphism 03C6. =
...
=
Proof of the application.
For
forward calculation that for all p, q,
Vol. 3, n° 2-1986.
= ax ~ ax~ it follows from r
a
strai g ht-
16
P. BONCKAERT
We see that for each h > 2 the linear map natrix with respect to the basis
Hence Im [X 1, A basis for Ker
[X 1, - ]h
has
a
diagonal
[Xi, -~ Hh. [Xi, 2014 ] h is obviously Ker
=
So, applying the normal form theorem,
we
have for
4>*X
=
X’
an oo
jet
of the form
It suffices to prove that § can be chosen in such a way that it formally preserves the z-axis. Because then X’ also leaves the z-axis formally inva-
riant, and hence
cannot contain pure z-terms.
component the Y’ E Hi with
Take any [X 1, Y’== bi. If we separate the pure zi terms of o a the and components of Y’, we can write it in the form:
-
-
where Y E HI has
no
pure zi terms in
itsax~ and a components. We have ~y
.
d
as bi and [Xi, Y]don’t contain pure z1 terms in their 2014 component, ax necessarily A 0. So bi [Xl’ Y ]. As ~1 = ~ -1) (the time -1 mapping of Y) we obtain the result. 0 In chapter IV we will make extensively use of the following result. But
=
( 1. 22)
THEOREM
=
(Brouwer, Leray-Schauder-Tychonoff fixed point theoAnnales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
rem) [Sm].
-
Let E be
a
locally
convex
[R~
117
topological
vector space, K a
nonvoid compact convex subset of E, f any continuous map of K into K. Then f admits at least one fixed point in K, that is : a point x E K with
/(.Y) =
~.
§ 2. One
The main result.
pose the
following question: suppose that a (generalized) formally invariant under a germ X E G3 (by formally we mean : up to the level of oo jets, or equivalently : up to the level of formal Taylor series); does there exist a Coo invariant one-dimensional manifold having oo contact with D ? The answer is yes, provided X is non-flat along D. This is one of the major ingredients of the following theorem. Let us emphasize that the results of this theorem are stated in terms can
direction D is
of germs in 0 E f~3. Let X E G3 and let D be a direction. If X is non-flat (2 .1 ) THEOREM. then there exists a Coo cone K of finite contact around D such that along D, one of the following situations occurs: -
I. all the orbits of X
hitting
K enter K and leave
K, except (of course)
{(0.0.0)~ II. D is
invariant under X ; there exists a direction D’ in K, D, which is invariant under X; if we arrange (by of X if necessary) that the orbit of X in D’ tends to 0
formally
contact with
having ~ changing the sign then either
A the only orbit of X in K tending to 0 is contained in D’ ; all the other orbits of X starting in K leave K ; B. all the orbits of X in K tend to 0 ; if we add 0 to such an orbit, we obtain a direction which has oo contact with D ; C. there exists a unique C° 2-dimensional subcone S of K such that
i ) S is invariant under X ; ii) all the orbits starting in S tend
to 0; if we add 0 to such obtain a direction which has oo contact with D; iii ) all the orbits of X starting in KBS leave K.
an
orbit,
we
Always assuming that along the invariant directions the orbits tend to 0 for t -~ oo, we find in cases II. A and II. C a unique model up to C° equivalence and in case II. B even up to C° conjugacy. The proof of this theorem will be given in chapters IV and V. (2.2)
PICTURES of the situations
Vol. 3. n° 2-1986.
occuring
in the theorem.
118
’. BONCKAERT
(2.3) re
REMARK. -
Perhaps sight.
in
some cases
the claimed results in
not evident at first
Take for
example
0
the linear vector field
1
(2.1)
0
+ -z2014 and X=x~x +y ~y time and have
) z-axis. All the orbits tend to 0 in negative ontact with the z-axis. Nevertheless one can find =
a
parabolic
of finite contact wo in this case) such that alle the orbits leave it, except of course the ne contained in the z-axis. This example indicates in which way the main leorem must be considered: once an orbit leaves the cone, we have no irther information about its future, which may be for example to tend ) zero or even to re-enter the cone. a cone
REMARK. - One can formulate and prove an analogon of theo(2 .1 ) in dimension 2, this time of course without situation II. C. On 1e other hand it is not yet clear to me how to generalize the theorem ) arbitrary dimensions; more specifically one could ask: does a formally mariant direction always hide a « real life » invariant direction? The mie question can be posed for diffeomorphisms.
(2.4)
m
Annales de l’Institut Henri Poincaré - Analyse
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VECTOR FIELDS ON
I~3
119
II. PREPARATORY CALCULATIONS CONCERNING BLOWING UP OF VECTOR FIELDS
Although the blowing up method for singularities of vector fields is known technique [Ta, Dul, Du2, D.R.R., Go ] we recall it here because it is of fundamental importance in the sequel. Especially for blowing up in a direction some specific calculations will be elaborated in full detail. Roughly spoken, blowing up a vector field is : write it down in spherical coordinates and divide the result as much as possible by a power of r (= Euclidean norm of x) such that one can take the limit for r - 0. a
§ NOTATIONS.
(1.1)
1. One We
-
spherical blowing
up.
denote, for m~N : 1m -+-1
So in fact with 03A6
we
introduced
Rm + 1 ) ( 0 }
on
a
sort of
spherical
coor-
dinates.
(1.3) PROPOSITION with X(0) 0. Then there is a Cp -1 field X =
on
Sm
If X
(or : ~~X = X). has flatness k, that is,
R such that in each q
x
C,(X(~)) =
e
S""
x
D =
0 and
~+iX(0) ~ 0,
then for the
following vector field -
we can on
S"’
take the limit tor
which
we
r
-
0 and
1 -
we
obtain
X is called the ( 1. 4) DEFINITION. X and up (once) diriding bij ~. -
Vol. 3, n° 2-1986.
a vector
held ot class
still denote X. vector field obtained
by blowing
120
?. BONCKAERT
X restricted to Sm x { 0} is a vector field tangent to Sm x {0}. The study of this vector field sometimes gives information on the asymptotic behavior of the orbits when they tend to 0 [Go ].
§ 2. One directional blowing
up.
In high dimensions the explicit calculations for spherical blowing up quickly become complicated. If we restrict our attention to one open halfsphere of S’" we use the better computable directional blowing-up. We can assume that the half-sphere has (0, 0, 0,1) as « north-pole ». x R by E Let us replace where E is an arbitrary normed vector...,
space. We do this because later on construction remains the same.
point
{0}
shall
use
this form and after all the
CONSTRUCTION.
(2.1) A
we
R.
x
(x, z)
~
of E will be denoted (x, z). With « the z-axis » we mean be the following map : T": E x R - E x ~: Let, for n E N, z). Similarly to proposition (1. 3) one proves
(2 .1.1 ) PROPOSITION. - Let X be a CP vector field with X(0) 0. Then there is a Cp-1 vector field X~1 on E such that in each q E E x ~ : if X(~’ 1 (q)) (or: X). Again jkX(0) 0 and we X 1 devide we denote the jk+1X(0) ~ 0, by resulting vector field =
=
X1
and
=
z
vectorfield
on
again we can
take the limit for
z
-
0 to obtain
a
still denoted XB
E
X1 is called the vectorfield obtained by blowing (2.1.2) DEFINITION. up X (once) in the z-direction and dividing by zk. Like in § 1, E x ~ 0 ~ is -
invariant under
(2.2)
RELATION
[R m+1 Let
so we can
consider the restriction
BETWEEN DIRECTIONAL AND SPHERICAL BLOWING UP IN THE
CASE.
denote the
insider the
ne
XB
«
upper
bijection
F:
hemisphere », that is : ST x R - Rm+ 1:i
immediately checks that ~ _ ~ 1
o
F.
Annales de l’Institut Henri Poincaré -
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121
VECTOR FIELDS ON
Hence X
If
we
As
~Sm X ~
denote
and X1
F(xl ,
1
are
C~’
k is
(1+
...
r)
Z,
... ,
conjugated : FX =
an
... ,
xrn,
CALCULATIONS
(X 1,
... ,
Xm, Z) then,
the orbits
+ same
orientation.
CONCERNING ONE DIRECTIONAL BLOWING UP.
GENERAL FORMULAS.
(2 . 3 .1 )
=
X1.
everywhere positive function,
ofX~and F~X coincide. They also have the
(2.3)
r
=
write X (Xx, X~ ). It is an easy calculation to
If X is
-
a
vector field on E
x
[R
we
=
see
that
X if only if
/~
,
and thus rrom
now on we assume
(2 . 3 . 2) some
1HE
00 JET OF
X to be 1
X IN U
(CASE {Rm + 1).
(usual) multiindex abbreviations. m
For
then
we
denote
We write the oo
jet
ot X in U down as
if X has flatness k in 0. Vol. 3,~2-1986.
-
Let
us
indicate here
122
BONCKAERT
’or the
oo
jet of X1
Jsing (2 . 3 .1 ) i) a" -
and
in 0
we
write
calculating straightforward ,...,03B1m)
we
find :
for 1
m 1 _ x
i
for 1 i m 0 ii) a for iii) ~ 0 I n; i i ), ii), iii ) we may take the right hand side zero whenever it is undefined. =
=
(2. 3 . 3)
REMARK. - In
nd for each
a
with |03B1|=
in (iii) we see that for each 0. This expresses the fact that the
particular n :
c:
=
is invariant under X1or equivalently that f X 1 does not contain pure xa terms.
yperplane
§ 3. Successive directional blowing
the ~
z
=
0
component
ups.
~.1) Construction.
Let X1 be obtained by blowing up X in the z-direction and dividing by zk. Suppose that X 1 has again a singularity say in (a, 0) E E x { 0 } and that ~ 1 (a~ 0) = ~)_~ 0. Then we can blow up X 1 in (a, 0) in the z-direction in the following itural sense. Denote T : E x R - E x (x, z) - (x - a, z) ; since T*X is a singularity of flatness p in 0, we can consider the vector field T*X1 stained by blowing up T*X1 in the z-direction and dividing by zp.
i
DEFINITION. up X twice in the
(3.1.1) owing
T*X1 is called the vector field obtained by z-direction first in (0,0) then in (a, 0) and dividing
-
by z~ then by zP; we denote it X2. Of course X2 depends on the choice of the singularity of Xl. In this way, ~ can of course go on with the construction and define successively X" in some singularity. Obviously Xn depends on the I blowing up oice of the singularities in which we blow up. Let us formalize the idea ving a prescribed sequence of singularities in which must be blown up. ’st
_
Let X be a Coo vector field on E (3 .1. 2) DEFINITION. ;0) 0. A directed sequence of successive blowing ups of X is a such that triples (X~, (xn, 0), u{ X° X and i) (xo, 0) = (0, 0) ;
with
-
=
sequence
=
Annales de l’Institut Henri Poineare -
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VECTOR FIELDS ON
0 _
ii )
N : Xn
n
has a singularity ot flatness kn m (xn, U) and
by blowing up X" in (xn, 0) and dividing by zkn.
is obtained
DEFINITION.
(3.1.3)
of X where dn, 0
A directed sequence of successive blowing ups N : (xn, 0) = (0,0) will for brevity be called a in 0 of X (in the z-direction). Such a sequence is
-
n
of blowing ups henceforth denoted by (Xn, 0, kn)o sequence
correspondence
n
N.
Further on we will show that there exists a 1-1 between directed sequences of successive blowing ups oo included). and N - 1 jets of directions (N
REMARK.
(3 .1. 4)
123
~3
-
=
(3.2) Calculations concerning
successive directional
blowing-ups.
We will give explicit formulas only for a sequence of blowing ups in 0. As we have already announced this is, at least for our purposes, no restriction. In this case we can define X~ in one step : LEMMA. 2014 Let Xn be obtained defined in (3.1).
(3.2.1) in 0 of X
by
a
sequence of
blowing
ups
as
== , ..,2014 ThenXnR. ~p~E
Xn where Xn has the property
=
X(03A8n(p)),
x
Proof
-
Straightforward.
D
(3.2.2) REMARK. - We see that properties of xn in a cylinder-shaped x [0, oo [ are transformed, by ~’n, to simular neighbourhood x [0, oo [. It is proporties of X in a cone of contact n -1 around { in this sense that we will obtain the results in the main theorem (1.2.1).
(3 . 2 . 3 )
FORMULAS. 2014 Just like in _
Further
on we
(3 . 2 . 4)
will need the
LEMMA.
/11
X
=
(Xx, Xz)
we
have
B
~~
following
Let X be
(2 . 3 .1 ) for
result.
Cx vector field on E x R. If X is non-flat x ~ is formally invariant, then there exists a Q E fvl and a C x function y: E with y(o, 0) ~ 0 such that + 1 the R-component of XQ + is of the form is the vector field z). obtained by blowing up X Q + 1 times in 0, in the z-direction without dividing by a power of z.)
along {OE}
x
[0,
-
oo
a
[ and if {
(XQ
Vol. 3, n° 2-1986.
.
124
P. BONCKAERT
Proof - Put X==(Xx, As, by our assumptions, the map z -~ z) has a nonzero jet, there exists a Q~N and C" maps Ao, Ai , ... , AQ : E ~ ~ with Ao(0) Al (0) == ... == A 1(0) - 0 and AQ(0) ~ 0 such that we can write =
Xz(x, z) Ao(x) + zA1(x) + The ~-component of XQ+ 1 is More explicitely : =
...
+
+
+
III. REDUCTION OF A SINGULARITY TO A SINGULARITY WITH NONZERO 1-JET BY SUCCESSIVE DIRECTIONAL BLOWING UPS
blowing up a singularity of flatness k one might hope that the atness of the resulting vector field in a singularity is not strictly bigger ian k. Up to one type of singularities, this will in fact be the case. If we consider a sequence of blowing ups (Xn, 0, we will see, Iso for that exceptional type of singularities, that after at most one step ve flatness kn becomes decreasing (perhaps constant). In fact, a vector eld which is the « blown up » of another one cannot be of that exceptional Ipe (mentioned above). Next we will prove that if the sequence kn does not decrease to zero, then ie vector field is flat along the z-axis. Let us start by showing that, up to a Coo change of coordinates, it is o restriction to assume that a directed sequence consists of blowing ups long the z-axis. When
§
1. Definitions and
properties
of directed sequences.
PROPOSITION. - Let N~N u { ~} and suppose that is a directed sequence of successive blowing ups of [hen there exists a C’ change of coordinates ~ : (p~m+ 1, 0) ich that in the new coordinates this sequence is ( ~,~(X)n, (0, 0),
0),
( 1.1 )
Proof
-
We define
inductively 1
on
as
some
1, 0) kn)o ~ ~ ~;.
transformations
follows.
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
!R 3
125
Put
Suppose
that ai,
’Ii
witn n + 1
defined tor 1
are
the singularity of ( ~n),~(X)n+ (~+i,0)e!R" (.Yn+i, 0) in the new coordinates induced by ~n’ Put
denote to
1
1
/
2014n~’ 1 -
/--
B
1
m.
We
corresponding ’B
1 ~mT 1 : (x, z) - (znx, z) denotes the « blowing up map », If qn : 1 N. n observe that ’Pn 0 Tn 03B1n o 03A8n, blown If Y E G~~ ~can be up n + 1 times in (0,0) and if ~n + 1 must be blown up in a singularity (an + 1, 0) then : =
.
~~
.
~
.
~ ~~
.
,
,
so must (the z component is not alterea by 1 n +1 nor by be blown up in (0, 0). we obtain that Applying the foregoing observation to 1 be is (which ( ~n + 1 )~c( 1 ) must blown up in (0, 0). For
(0)
’--
SO .
r or nmte
ØN -1
I
is tne aesirea
.
I
/~.....
i.,~
~).
by (1), consider the inverse limit of the ~", and theorem (I .1.12) of Borel together with the inverse function theorem [Di ] gives the desired ~. D For N
=
oo we
can,
For each directed sequence ( 1. 2) COROLLARY. (xn, 0), there exists a C~ change of coordinates (x, z) z’) and a direction D = ~ -1 (z-axis) such that if we blow up ~ * 1 X successively in 0 then points corresponding to (x, z) (0, 0) like in the proof of ( 1.1 ), are precisely (x’, z’) 0). Conversely, given a direction D = ~ -1 (z-axis), there exists a sequence ~ as above. Moreover, D is unique up to N -1-jet equivalence. In other words : there is a 1-1 correspondence between directed sequences and N -1 jets of directions. So we can speak of a sequence of blowing ups along a direction. -
==
=
==
Vol. 3. n° 2-1986.
126
.
BONCKAERT
If we blow up X successively along the z-axis it is of course possible hat after some finite time there is no singularity any more in (0, 0). Let us ~rmalize this as follows.
is called a maximal directed sequence 0, ( 1. 3) DEFINITION. >f blowing ups along the z-axis if either i) N oo ii) N EN -
=
_
Property. Such
a
sequence
always exists.
If (X",0,~)o~N is a maximal directed sequence PROPOSITION. vith N E N, then there exists a cone K of contact N - 2 around the z-axis ,uch that all orbits inside K enter K and leave K after a finite time.
( 1. 4)
-
Proof - As X~’~(O) 5~ 0, we can construct a cylinder shaped neighbourhood of 0 in [Rm x [0, oo [ of_the form C= {(~ R, : E [o, b [ } such that all orbits of XN - 1 inside C enter C and leave C after 1 finite time. Then
Let K be the germ in 0 of this set. D So from now on we will consider infinite sequences of blowing ups. Let us indicate what it means for X that a maximal directed sequence of blowing ups of it along the z-axis is infinite.
(1. 5)
PROPOSITION.
sequence of are
blowing equivalent :
Let ups of X -
along
be a maximal directed the z-axis. The following statements
i) N ii) the z-axis is formally invariant under X (see definition (1.1.20)). Proof i) ==> ii). Let us write down the oo jets as follows -
0 for all K > ko + 1 and i E Ve must prove that afo he formula in (II.2.3.2) ii) we find that for all K =
Annales de l’Institut Henri Poincaré -
>
ko
Analyse
+ 1 and non
linéaire
VECTOR FIELDS
i
1,....m}:
==
ON R3
and turther
aki,0
127
induction
by
on n we
get,
ato. using Suppose, by contradiction, that aKi,0 ~ 0. Then -k0-1 ~ 0 so k 1 + 1 K - ko - 1 (remember the choise of kin definition (II . 3 .1. 2)) 2 ; further by induction K - ko with always, as it should be, 0 kn _ 1 - n. kn _ 1 - n for some n. Certainly sooner or later 0 K - ko the
same
formula, for all n
=
...
=
-
...
-
m
X"(0) =
But then
ii)
~
i ).
ko diately ai,0 K >
Let
+ 1
~ 0, contradicting N
oo.
notations. We have ...,~1}. The formula (11.2.3.2) and f ~{ 1, ...,m} whence
the
us use
and 1, 0 for all n 0
=
the result.
=
§ 2. Reduction
to
same
a
singularity
with
nonzero
The main purpose of this section is to prove the
0, for all gives imme0, dn and =
1-jet.
following :
If is a maximal directed sequence (2 .1 ) THEOREM. of blowing ups of X E Gm+ 1 along the z-axis { 0 }m x [0, oo [ and if X is non-flat along the z-axis then there exists a N~N such that has flatness zero The proof of this theorem will be a consequence of the following propositions (2.2), (2.3), (2.4), (2.6), (2.7) and (2.8). D -
(2 . 2)
PROPOSITION.
-
Let X E Gm + 1 have flatness k. If X 1 has flatness n
As
0 ~x ~ k
=
1, 1
U we
i
~
only have
to look at the
a1 x 1
and the
1,
Out of (11.2.3.2) Üi) we obtain for each n k and a with = n -1: 0 with 1 _ n 1g = ci ~ ~ ~ ~ " ~ ~=c~ ~so~~ ~ =0, for each x with 0 ~ | 03B1| k. (2) From (I I . 2 . 3 . 2) i ) we get for each n with 1 n k + 1 and a with x = n and 1 xl : +
m.
=
But by 0 ~ x
(2) above those c’s ( _ k + 1 and 1
Vol. 3. n° 2-1986.
are zero.
xi.
Consequently
==
0 for each
oc
with
128
In the same way, using (11.2.3.2) ii), we obtain that all the coefficients in jk+ iX(0) vanish except possibly c~ ~1 with = k + 1. D
(2 . 3) PROPOSITION. has flatness
-
and if X~1 has flatness k, then X‘
If
k.
Proof Suppose that jk+ 1X2(4) that imply -
=
0. Then
proposition (2.2)
would
_
By our assumptions one of the ( = k + 1, must be different from zero. But this is impossible because of the remark (II. 2.3.3) (invariance of the z = 0 hyperplane). Q If XEGm+1 has flatness k and (2.4) PROPOSITION. then X 1 has flatness k + 1. -
Proof - We must show
and from jk+1X1(0)
=
0
we
that
~+2X~(0) 5~
0. From
proposition (2.2)
get
Because X has flatness k, there exists Using (II . 2 . 3 . 2) iii ) we observe that
an a
-k+2 - ~+2+~-~-1 _
# 0.
= k + 1 and
with |03B1| 1 .
D if (2 . 5) COROLLARY. - If sequence of blowing ups in 0 of X in the z-direction if X has flatness k then either.
is
(N
a
u{
directed oc }) and
i ) Xhas flatness smaller than or equal to k and for all n with 0 11 N -1I the flatness of Xn +is smaller than or equal to the flatness of Xn ; 1 ii) X has flatness k + 1 and for all n with 1 - n N - 1 the flatness of is smaller than or equal to the flatness of X~. Let X E Gm + 1. Suppose that there exists an (2 . 6) PROPOSITION. of blowing integer k > 1 and an infinite directed sequence (Xn, 0, -
X~ has flatness k and at each
ups in 0 of X in the z direction such that Vn
step
divide by zk. Denote i: R ~ Rm+1:z
we
-
(0,
...,
0, z).
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
is
II$3
129
Proof - Let us write again X for its formal part. Then jk- i(X)(0,..., 0, z) formally determined by
We
use
the notations
"
2014t~_2014
,
as
in
(II.2.3.2) and obtain 00
Hence the terms
n
playing
a
role in
o
i)(O)
are
1
we look at what these terms
give when blowmg up
We use qn:(Rm+1 ~ the formula in (II.3 .2) which becomes here m
For the
2014 component C,Y,
’
~
Vol. 3, n° 2-1986.
~--i
we
obtain
as In tne
proposition. and
30
a nd for
the - component we have
~ irst look at
(4). If this construction is possible for each
n E N then
Thus the second EE in (3) vanishes. For the same reasons as 0 for all N >- k + 1,0 z m, 0 - ~ tioned we must have Th is means 0 1 (X) f)(0) = 0.
c~
=
0
just men-
=
_
o
Let be non-flat along the z-axis. (2 . 7) PROPOSITION. Let xn be obtained from X by n successive blowing ups in 0 in the z direction. -
Then X~ is non-flat
along
the z-axis.
Let ~n : ~m + 1 ~ ~m + 1 ; (x 1 ~ ... , .xm ~ Z) -~ Proof z) the blowing up mapping leading to X~. Then for some analytic function Gn we have and X o ~I’n= ~~ X". Denote i : [0, oo[ ~- p~m + 1: z (0, ... , 0, z). Suppose that j~(Xn f)(0) 0. Then j~(Xn f)(0) 0 and hence j~(X f)(0) 0. But i = i. This would imply that j~(X i )(0) 0, contradicting 3ur assumptions. D -
...,
Je
o
-
o
0
=
=
0
0
=
o
=
Suppose that X E Gm+ 1 is non-flat along the (2 . 8) PROPOSITION. z-axis. Then it impossible that there exists an infinite directed sequence of blowing ups in 0 of X in the z-direction with Vn > 1: kn > 1. -
Proof - Suppose that there exists such a sequence. Then there exist integers k > 1 and N > 1 such that Vn > 0 : XN + n has flatness k. Hence proposition (2 . 6) ~)(0) = 0 contradicting proD position (2.7). This completes the proof of theorem (2 .1). o
IV. TREATMENT OF ALL THE FLATNESS ZERO CASES AND PROOF OF THE MAIN THEOREM, WITH EXCEPTION OF THE C° RESULTS
study germs in 0 of vectorfields which i ) leave the [R~ x { 0 } hyperplane invariant ii) leave the z-axis { formally invariant iii ) have a nonzero 1-jet. (Sometimes we will only consider the case m 2.)
We
=
Annales de l’Institut Henri Poincaré -
Analyse
non
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131
VECTOR FIELDS
The
1-jet of such
a
vector field has a rA
matrix of the form
AI
m
with A E f~’~ " and where Cô has the same meaning as m (11.2. 3 . 2). Such vector fields appear as a result of the reduction by successive blowing up in part III. Even if the 1-jet is nonzero, we will sometimes blow up the vector field some more times in order to obtain situations like in the main theorem (1.2.1); also in some cases the proof of the existence of the Cx 1-dimensional invariant manifold is made easier by making extra blowing ups.
We must show that all possibilities for C6 and A (both not simultaneously a situation as in the main theorem (I .2.1). 0. This turns out to be fairly easy. We first distinguish in § 1 the case If C6 0, (§ 2) we restrict ourselves to the case m 2. The possibilities are :
zero) lead to =
=
i ) A is hyperbolic : see (2 .1) ii) A has eigenvalues ( « the rotation case » ) : see (2. 2) iii ) A has eigenvalues a, 0, a E tRB { 0 }: see (2 . 3) iv) both eigenvalues of A are zero : see (2 . 4). .
§
1. The
eigenvalues
in the z-direction is
nonzero.
satisfies : Suppose that ( 1.1 ) PROPOSITION. is formally invariant under X i ) the z-axis { 0 (see the notations of (11.2.3.2) or above). ii) -
Then
a) there exists a Coo germ h : ([0, oo [, 0) 0) whose graph (germ) I z E [0, oo [} is invariant under X and with j~h(0) 0 ; { (h(z), z) E Rm R b) supposing c10 0 change the sign of X if necessary there exists a cone K of finite contact around the z-direction x [0, oo [ such that the only orbit of X in K tending to 0 is contained in {(h(z), Z)E ~m ze [0, oo [} ; -~
=
all the other orbits
Proof ko ki1 =
a
starting in K leave K after a finite amount of time. If we use the formulas in (II.3.2.3) (with of course ... = kn-1 0) we find that for all n > 1 the 1-jet of X’~ has
-
-
=
matrix of the form
() l
Notice that 5. is
Vol. 3, n° 2-1986.
an
eigenvalue of A
if and
only
if
is
an
eigenvalue
132
P.
BONCKAERT
Hence for large n, X~ is a hyperbolic saddle for which we can apply the (un-) stable manifold theorem (see for example [H.P.S., Ke ]) to obtain the results. D
(1.2) REMARK. - The methods we develop further on for the delicate situations would also work here; the calculations would be much simpler here.
( 1. 3)
REMARK. - In
§ 2. From
(2.1)
The
( 1.1 )
eigenvalue
~3,
work in
now on we
THE HYPERBOLIC
we
may
replace
~
by
in the z-direction is
more even
any Banach space.
zero.
thus: m = 2.
CASE.
(2.1.1) PROPOSITION. Suppose X E G3 satisfies z-axis the is i) formally invariant under X -
3
8
iii )
a, b > 0
by
ii) X is non-flat
Then there exists
along the a cone
z-axis.
K of finite contact around the z-axis, a a COO germ h : ( [0, oo [, 0) -
C°2-dimensional subcone S of K and such
a) b) c)
that j~(0)
=
unique (fF~2, 0)
0 and :
S is invariant under X the graph of h, ~ (h(z), z)z E [0, oo [ }, is invariant under X and lies in S in K we have situation II. C of the main theorem (1.2.1).
8
Proof - By
lemma
(II.3.2.4)
we
may
assume
that the
- component
of X is of the form zQy(x, y, z) with y(0, 0, 0) # 0 and Q > 2. We may, and do, assume that y(0, 0, 0) > 0, because if not, replace X by - X which is of the same type. If we use the center manifold theory such as in [H.P.S., Ke] we obtain the existence of a unique center-unstable-manifold. Although it is general not necessarily CX, here we can use the uniqueness of the center-unstable manifold as well as the fact that X is non flat along the z-axis to obtain that the center-unstable manifold is in this case, as follows. For each In E N there exists a C7" center-unstable manifold Wm defined on some neighbourhood Um of 0 ; we can take care that U m +1 c Um for all By uniqueness of the center-unstable manifold we have Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
1
If
=
holds for fixed
we
take U 1 small
.
enough
then the
133
following
m.
Since ~(0,0,0)
>
0 and since a
>
0 there is a T > 0 such that
denotes the flow of X). So Ui1 n Wi is a Cm manifold. But as m is arbitrary, we obtain that U1 n Wi is C ~. The behavior of X restricted to this center-unstable manifold found in full detail in [D.R.R. ]. D
(4)x
can
be
The methods to obtain invariant manifolds, deve(2.1.2) REMARK. further on for the more delicate cases, could also be applied here. lopped -
(2 .1. 3) defined
PROPOSITION.
-
Let E be
Banach space, X
a
a
Coo vector field
with X(0) 0. Suppose that neighbourhood U of 0 E E the z-axis is invariant under i) X; formally is a hyperbolic contracii) E x { 0 } is invariant for X and D(X IE tion or expansion, that is : the spectrum of this linear operator is contained on a
=
x
in a subset of C of the form { z for some i~ e R, h > 0 ; iii ) X is non flat along the z-axis. iv ) X is bounded on U together with all its derivatives. Then there exists a cone K of finite contact around the z-axis and a C ~ germ h : ( [0, cxJ [, 0) - (E, 0) with a) the graph of h, ~ (h(z), z) z E [0, oo [ }, is invariant under X b) in K we have situation II. A or II. B of the main theorem (1.2.1).
Proof - Let us write X (Xx, XZ). By lemma (II . 3 . 2 . 4) we may assume =
’2014 ~- ~
,
0 and Q >_ 2. There exist A e Lc(E, E) and B: (E x R, 0) - (E, 0) such that
that
A,/~
XZ is of the form : ,
with 7(0, 0) #
and such that
(E
x
~, 0)
-
(Lc(E, E), 0) and Rx:
(up to changing the sign of X).
r Re z i ) the spectrum of A lies in { 0} ii ) B and Rx are Cx germs; especially B(O,O) 0 0) = 0. iii) =
is
This because the z-axis is formally invariant. We also may assume that R x flat along E x { 0 } because if not, blow up once.
Vol. 3, n° 2-1986.
x
134
P. BONCKAERT
The existence of the invariant graph follows from the center manifold Its smoothness follows from the fact that y(0, 0) # 0. If y(0, 0) > 0 then we have situation II. A and if y(0, 0) 0 then we have situation II. B; this can be proved by standard techniques like in [D.R.R. ] or like further on in the more delicate cases. D
theory.
(2.2)
THE
ROTATION CASE.
We study germs in 0 of vector fields on R3 which leave the z-axis R2 0 formally invariant and for which the 1-jet has eigenvalues here E Such a vector field is (/. fRB { 0 } ). completely non-hyperbolic, so the « classical » theorems about invariant manifolds as in [Ke, H.P.S. ] are not applicable. First of all, such a vector field has, up to a linear change of coordinates preserving x {0} and {0}2 x R, a 1-jet of the form -
In
[B.D.]] we obtained
the
(2. 2 .1) PROPOSITION. i ) the z-axis { 0}2 /
iii)
A is non nai
x
along
-
following
result :
Suppose X E G3 satisfies is
formally
invariant under X
x 1
me z-axis
then
a) there exists a Coo germ h : ( [0, oo [, 0) (~2, 0) whose graph (germ) { (h(z), z) e [0, oo [} is invariant under X and with jooh(O) 0 b) there exists a cone K of finite contact around { 0}2 x [0, oo [ such that in K we have situation II. A or II. B of the main theorem (1.2.1). -
=
(2.3)
ONLY
ONE NONZERO EIGENVALUE.
Here we will meet all situations II.A, II. B, II. C of the main theorem. In situation II. C we will have to construct invariant surfaces. Sometimes we could make use of the classical center manifold theory, but for other cases, where the requested invariant manifold is not a stable, center-stable, center, center-unstable or unstable manifold, we will have to apply our own methods. But if we set up the whole machinery anyway it can effortlessly be applied to the « classical » cases. The methods to obtain smooth invariant manifolds developped in this section, can also be used to obtain the results in (2.1) and (2.2). Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
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135
Up to a linear change of coordinates preserving (~2 x ~ 0 ~ and { O}2 may assume that the 1-jet is of the form
x
R,
we
(2 . 3 .1)
PROPOSITION.
the z-axis is
i)
-
formally
Suppose X E G3 satisfies invariant under X
a
ii)
= ax
a ~ 0
X is non-flat
iii )
then there exists a germ h : ([0, oo [, 0)
along
the z-axis
K of finite contact around the z-axis and (~2, 0) with /~(0) 0 such that
cone -
a
Coo
=
a) the graph (germ) ~ (h(z), z) z E [0, oo [} is invariant under X b) in K we have situation II. A or II. B or II. C of the main theorem (1.2.1).
Proof. - We may assume that Because X is non-flat along the
z-axis,
and hence we assume that there exists of X is of the form
with
fl/(0, 0, 0) #
So X
can
in (I.1.21). apply lemma (II.3.2.4)
has the form
a
we can
Q e
as
N such that
the ~ component oz
O.
be written
as
follows:
where
0 i) fl~~~ 0) == ii) 0, o) ~ 0 i) S , is a vector field which is
oo
flat
along
the
z
=
0
plane
and with
c
zero
-,--component c~
(we absorbed
it in 7;
we
also assumed
at
least
one
blowing up) it;) Q > 2. We only pay attention Vol.
3,
n° 2-1986.
to the upper half space
[R2
x
[0,
oo
[. If we blow
136
P. BONCKAERT
up X n times, without dividing is nonzero, then we get
by
a
power of
z
of
course
since the
1-jet
;=)
0 plane. for some S~ which is oo flat along the z Because y(0, 0, 0) ~ 0 we can assume that X", denoted again X, has the following expression (new fl , y, ... ) provided that n is big enough : =
~-) /,(0,0,0)=0 it) /(0, 0,0) = ~ ~
0
P E { 1, ..., Q - 1}
~
is (X) flat along the z 0 plane and has zero ic) component. ~ V) y(0,0,0)=c~0. Furthermore, in order to make the calculations a bit simpler, we divide X ==
by
1 1 + -
The
a -
/
resulting
vector
-
field, again denoted X, is of the form
,
(new f; ... ) :
where f, y, S x satisfy the same properties (ff) to (v) as above. The foregoing manipulation is not essential. If we look back to the expression for X~ hereabove, we see that in case P Q - 1 we may assume that b. c 0 provided n is chosen big enough. So up to a change of the sign of X the four following situations can occur : =
I. II. III. IV. Annales de l’Institut Henri Poincaré -
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137
VECTOR FIELDS
Here
we
already
theorem, II treat
that I will lead to situation II. B of the main situation II. C and IV to II. C. We We will refer to some lemmas which will be
announce
to situation II. A, III to
each case separately. after this proposition.
proved
In order to detect invariant graphs for X we consider the graph transformation defined by the time one mapping of X (precise definitions will follow). Roughly spoken this is: take a graph of a map (x, y) h(z) and transform it by the time one mapping of X. First we modify X, without modifying its germ in (0,0,0), with a Coo =
« bump » function r : [0, on [1, oo [ as follows. For all 8
>
0
we
oo
[
-
[0,1 ], where i(u)
=
1
on
-0, -
-
and z(u)
=
0
define / 7X
Then XG has the same germ in 0 as X can be defined on a neighbourhood of the form (x, [0,00[, with B(0~)== { (x, y) E }. Moreover the z)| z ~ 8 } is invariant under - X~. We can modify X to XG in such a way that the (new) f, y satisfy satisfy inequalities like ~
-
r,__
-.
~B
~
~
-
A
neighbourhood B(o, ,u) x [o, oo [, where al, a2, bl, b2 can be independent of E (for this reason we don’t attach an index E to f or y).
on some
chosen Let
us
show that the
«
fiatness condition
»
of the
is independent of E, that is : for all r, S E N there exist b such that for all E > 0 and for all (x, y, z) E B(o, ~) x 1
Let in tact r,
Vol. 3. n° 2-1986.
,.
i
i v
W e have :
, ,
E
term 03C4(z ) >
S x (x, y, z)
0 and
[o, 3] :
>
0
138
P. BONCKAERT
There exists a ð > 0 such that for all p E ~ 0, there exists a constant Li,p > 0 such that
...,
,
- ~ --
z
The time
1-£)= mapping
~ ~:
one
z) =
(1
0
+ s ~ and all i E { 0, B(o, /1) x [o, ~ ] :
...,
s}
,
Furthermore for all j e fil there exists I T~(M)) ~ K~ . Hence for all z 6:
and for all
r
on
so
the
of xE
can
+
y,
a
K~
>
0 such that for all
[0, oo [:
Me
problem is trivial there. be written down
z))y, z
+
y,
as :
z))
+
y,
z)
where
i)
is oo-flat
along [R2 x { 0 }
and the flatness condition is
indepen-
dent of 8
has zero z-component (we absorbed its z-component in g) ; ii ) iii ) there exist constants b2 and a neighbourhood
independent
of ~, such that
on
V:
We write Fe We will show in lemma (2.3.5) that there (Fx,e, Fy,e, > 0 and 03B4 E ]0, 5i[such that if h : [0,5]] - [R2 satisfies h(0) 0 and for all ze [0~]: [ h’(z) ( [ 1 then Z : [0, 5]] z z) is a diffeomorphism onto (at least) [0, ðl]. Let z(Z) denote the inverse. We define, for 0 8 61, the following =
exist 03B411
=
function spaces :
[0, ð] -+ [R2
F?,. = ~ h
is Cm,
h( [~~ ~ ])
c
B(0,
1, h ~ - o
and for all
[0,5]]
-
[R2 is Cm, h(0)
I Define H
=
as
H(Z) =
=
0} [s,5]: ~(z)==0} i E ~ 0, m~~. ze
[a,3]: h(z)
=
0 and for all z E
[
for all
z(Z)),
...,
z(Z))) if Ze[0, ð1]]
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
ana
=
U lI L E
L~$
139
is a wen defined
1E
map
iium
into
tsE-
is small and Fg is continuous for the C"" topology. From lemma (2.3.6) we will obtain that for all m > 1 there exists an c Fm. E > 0 such that for the Let Fm be the closure of F7 in topology. Each Dth for h E Fm and Lipschitz function with a Lipschitz constant independent of h ; this is still true for h E Fm. The space Fm is hence compact for the topology, as a consequence a theorem of Ascoli-Arzela [Di ]. of F~ is also convex. =~ As F is continuous on F~ for the topology, we still have
provided 8
Fr. Hence it follows from the Leray-Schauder-Tychonov fixed point theorem (1.1.22) that F has at least one fixed point in F~. We hence obtain for each m an Em > 0 and a C"" map hm: [0, 6 ] - [R2 with jmhm(0) 0 whose graph is invariant under =
z) (0, 0, 0) hm(z) 0 on [~m, 03B4 ] and because lim in z the estimates we is on obtain ceo outside 0. uniformly (see g) that hm In lemma (2 . 3 . 7) we will prove that if h : [0, 5 ] is C1 on [0, 5] B(0, p) and COO on ]0, 5]] and if the graph of h is invariant under then h is ceo 0. on [0,5]and jh(O) Hence hm will be So we can consider for example F~and hi. As a matter of fact we want hl to be invariant under XE1. To see this, observe that for all t E [0, 1 ] : ~xEl((0, £5), t) lies on the z-axis denotes the flow of Xe1) since ~1 5i; for all t’ > 0 we write t’ = n + t where t E [0,1 ] Because
=
=
-
=
and n
we
have I
~~~
C~B
t
tB
/
t
///1
C*B
B B
B
nence me
grapn 01 ni is invariant under 03C6x~1(.,J tor an t’ > u. in Finally lemma (2 . 3 . 8) we show that in some neighbourhood of(0,0,0) all orbits tend to (0,0,0) and that those outside [R2 x { 0 } (in the blown up situation) are graphs of maps h satisfying the hypothesis of lemma
(2.3.7). Hence
we are
in situation II. B of the main theorem
Here the graph transformation can be defined mapping of X is again of the form
where
is oo tlat
Vol. 3. n° 2-1986.
along
the
z
=
U
plane
as
(1.2.1).
follows. The time
and has a zero
one
z-component.
140
P. BONCKAERT
and
This time there exist constants (0, 0, 0) such that on V we have ~-
~
Here
we I" -
-
_
0 is
8 >
_
-
_
(Fx, Fy, sufficiently
.,
_
~m
~
"
,
n
--~-
small and if h E
~
then the map
(at least) [0, e]. To
~
~
.
~
prove
D20141/7/ B B .
this, remark first that
/
estimate the first order derivatives of of h E F~. Second observe that for small
we can
independent
Denote
~.~
=
is a diffeomorphism onto for small z:
because
..
V of
take the function spaces
Denote F
If
../~
neighbourhood
a
the inverse of Z ; put H
rtB
g and
h
by constants
s:
I1 h where
H(Z) = (Fx(h(z), z), Fy(h(z), z)). Again : i ) r : Fm B~ is well defined if 8 is small and r is continuous for the C"’ topology c ii ) for all m > 1 there exists an 8 > 0 such that proof see lemma (2. 3. 9) hereafter iii ) r has a fixed point in Fm (the closure of Fm for the topology). z
=
-
_
In order to prove that hQ is Cx we will show in lemma_(2.3.10) that there exist > 0 such that is a sequence in B(0, p) x [0, s] ~Z, with lim and for i ~ 1 (0, 0, 0) (xi-1, Yi- 1, y,, z,) y,, z; ) then this sequence must lie on the graph on hQ. This implies that for all m > Q there exists an ~’m > 0 with ~’m ~ min { 8m, 8Q } such that h,~ and hQ coincide on [0, s~ ]. As a consequence of the movement of F in the z-direction (see the estimates on g above) we see that hQ is C7" 0. Hence hQ is Cy and jx h(0) on [0, GQ]] and that 0. Because of the unicity of hQ. in the sense of lemma (2.3.10), we see as follows that the graph of hQ is also invariant under X. Let q ~ N/{0}. Then, by similar arguments as above, we find an E E ]0, 8Q [ and a C x map h : =
==
=
=
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS
[0, I ]
---
fR2 with jx h(o)
(the time 1 q must.
by
mapping
lemma
of
(? . 3 . 10),
=
0 whose
X). lie
Take
on
the
ON R3
141
graph is invariant under 03C6x(1 q,.) z E
[0, ~].
Then the sequence
graph of ,
(l )
So h hQ z ] . Hence the graph of hQ is invariant under ] q / for each q. Thus under X.. lemma it will II. A from follow that we in situation are (2 . 3 .11) Finally of the main theorem. ==
Case
I I I : P Q - 1, ~ 0, b > 0,
,
c > 0.
Let us first search for an invariant surface tangent to the yz-plane. This will be a so-called center manifold. If we would apply the « classical » center manifold theorem, we would obtain for each r the existence of a C7 center manifold : see for example [Car, Gu, H.P.S., Ke, M.M. ]. But we will show that in this particular case the center manifold is Coo, at least in a blown up situation. Further we will show that the orbits starting in some neighbourhood of 0, but starting outside the invariant surface, will leave the neighbourhood. The behavior of X restricted to this invariant 2-dimensional manifold tangent to the yz-plane is well known and is as follows: there exists an invariant direction D’ having oo contact with the z-axis ; all the orbits starting in the invariant surface tend to 0 in negative time and have oo contact with D’. So we will be in situation II. C of the main theorem. But let us now come to the proof of all this. First we prove the existence of the invariant ex surface. For reasons which will become clear later in the lemmas we first perform a rescaling of the z-axis as follows. We want in fact that P > 2. Let R : [R2 x ]0, oo [ - [R2 x ]0, oc [: (x, y, u) - (x, y. u2). We have that R*X’ == X if and only if for
Vol. 3. n° 2-1986.
142
Here in
P. BONCKAERT
particular :
instead 01 M, A instead 01 X .
Let us write again
z
We obtain
vector
again
a
field of the form
(new P, Q,f, ...). ~
~
~
with this time P > 2 and still P Q - 1. Let us already remark here that such a rescaling will have no influence on our result, since it is our aim to construct an invariant graph of a Coo function z) with for all y: jxh(y, 0)=0: it is easy to show that then also a C x function of (x, u) oo-flat along the y-axis. Also all the other results we shall obtain are preserved by this rescaling. The time one mapping of X can be written as
~==/!(~~/M) is
F(x, y, z) = (eax, (1 + where a(0, 0, 0) > 0, g(0, 0, 0)
z))y, z + zQg(x, y, z)) + 0, Roo is oo flat along the z
>
=
has zero z-component. There exist constants al, a2, such that on V:
bi, b2 and
n / n
We look for
a
Ceo invariant ,.
~..
,
,
*~
>
graph ...
I
,
,
’7
a *~
z) 0 plane and R
neighbourhood
V of
~
(0, 0, 0)
n-
of the form -
,
,,
> 0 will be rechosen some times for our purposes. If , 3 > 0 c V. small then VÓ,Jl:= [- /1, p] x [-~/~]] x [0,5] ci V and Let us define some function spaces for ~ ~ 3 :
The
are
and FI = Fx, F2 Let us write F (Fx, Fy, (Fy, F,). We would like that the surface determined by ( y, z) - F(h( y, z), y, z) is again the graph of some function. For that purpose we shall prove in lemma (2.3.12) =
=
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VECTOR FIELDS ON
143
[R~
sjJ small /1, 8 > 0 and h~F10,~ the map (Y, Z) : [- , ] [0, onto a is ~c ] x [0,8]. /1, (at least) [ diffeomorphism F2(h( y, z), y, z) ( y, z) In that case we can take the inverse of (Y, Z), and denote it ( y, z). So ( y, z) is defined on (at least) [ - ~c, ~c] x We put H rh where -
that for
-
=
H is defined on (at least) [ - /1, /1]] x [0, e]. So, for 8 small, r is a map defined on and takes values in B~. r is Moreover continuous for the em topology. In lemma As Fm c for small 8, r is defined and continuous on we that there a shall exists > that 0 such for each (2. 3 .13) prove (fixed) ~ > 1, there exists an 8 > 0 such that c Fm. From this fact, from the theorem of Ascoli-Arzela and from the Leray-
Schauder-Tychonoff fixed point theorem (I.1.22) we derive, in an analoguous way as in case I, that for all m E N there exists an 8m > 0 and a C"" map R with jmhm(y, 0) = o for all YE [ - , ] whose graph hm : [- , ] [0,~m ] is invariant under F. In lemma (2. 3 .14) we shall show that there exists 0 such that if (xi, yi, is a sequence in [ -,u, ,u] x [ /1, /1] x [0, 8] with and F(x;, yi, 1 ) then this sequence must lie on the graph of hQ. Again this uniqueness of the invariant graph and the movement in the z-direction (see the estimates on g) imply that hQ is in fact ceo and that its graph is invariant under the vector field X. Also j~hQ(y, 0) 0 for all y E [ - ,u, ]. -
-
=
Next we consider the restriction of X to this invariant C°° 2-dimensional manifold obtained above (always restricted to the upper half space). This 2-dimensional situation is treated in full detail in [D.R.R. ], and is hence omitted. Finally, from the expression of X we can now easily show that we are in situation II. C of the main theorem (1.2.1).
Here
we
also look first tor
an
invariant
surface, this time tangent
to
the xz-plane and passing through the x-axis. In contrast with case III this surface is not a center manifold. But nevertheless we will prove that we are in situation II. C of the main theorem, that is: the orbits starting outside the invariant surface and starting in some small neighbourhood of 0 leave this neighbourhood, on the other hand the orbits in the invariant surface just mentioned tend to 0 in negative time and have Jo contact with the z-axis. One can observe in the sequel that in this case it is crucial that the xz-plane is formally invariant, thanks to the normal form. If this were not the case, our method wouldn’t work. Vol. 3, n° 2-1986.
144
P. BONCKAERT
The time one mapping F of X has the same form as in case III with this time (x(0, 0, 0) 0 and g(0, 0, 0) > 0. Hence there exists constants al, a2, bl, b2 and a neighbourhood V such that on V : 17
If
0 are small then F(V,,,) V. We define for E 3 :
/1, ð
>
V J-l,ð
~==
[- /1, /1]]
x
[-
]
x
[0, (5]c
V and
c
Again: see i ) for small 8 we can define the graph transformation r on lemma (2. 3.15) here after. B~ is continuous for the Cm topology; ii ) there a (fixed) ,u > 0 such that for each m E exists >- 1, there iii ) exists an 8 > 0 such that Fm : see lemma (2 . 3 .16) ; iv) applying the theorems of Ascoli-Arzela and of Leray-Schauderthe existence of an 8m and a C"’ map hm : Tychonoff yields for all m 0 for all x E [ - ~c, ,u] whose [2014~,~]] x [o, Em ] ~ R with graph is invariant under F; v) the uniqueness in the sense of lemma (2.3.17), and the movement of F in the z-direction (see the estimates on g) imply that hQ is in fact Cx and invariant under the vector field X. 0) 0, for all x E [ 2014 ~ ,u] ; vi ) the results of [D.R.R.]] concerning the behavior of X restricted to the invariant surface together with lemma (2.3.18) hereafter imply that we are in situation II. C of the main theorem. D =
=
Lernmas used in the
proof of proposition (2 . 3 ..1 ),
(? . 3 . 5) LEMMA. 2014 There exist 6 > B(0, p) satisfies h(0) 0 and !! [0, 6] -
0 and
case
I.
a 1 E ]0,5[[ such that if h : on [0, 5]then for all 8 > 0
1
=
the map v.
is a
diffeomorphism
1
_
onto at least
FTD . -,
. 17
/ L/ ~B-B
[0, 03B41J
Annales de l’Institut Henri Poincaré -
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ON R3
VECTOR FIELDS
Further Z’(z) > ð11 = 5 + D (independent of h). Put
LEMMA.
(2 . 3 . 6) such that
Proof we
+
0(zQ)
>_ 21 for small ð
1, there exists
an 8 E
]0,~i[[
Fm.
c
-
QzQ-1b1
1 +
For each
-
145
Let h E
Fm and put
H
=
reh.
For
z
=
z(Z) and (x, y)
=
h(z)
have
0 and P Q - 1 and a 0. induction for all on i, . that Suppose, by II and Let us abbreviate ... , F1 h(z) := (h(z),z ) :=(Fx~E, j E ~0, 1, 2014 1 }. and F2 :== times and obtain for We differentiate the equality the left hand side, using the higher order chain rule [A.R., Ya ] :
since a2
_
_
where
is some « universal summation over
C10)
properties jl
+
+
...
jk
isolate the term with k :>
(F2 h))(z) :>
=
=
i and
jp
>
1 for all
WIth tne
Let
us
i in the summation : :>
=
...,7~
/?e { 1, ...,~c}.
h)~Z)~
...,
i- 1
There exist C x functions
D’(H (F2 ~))(’) ç
0
=
Ak, k E ~ 0,1, ...,i DIH(FZ(h(Z))) - (D(F2
ç
;-1
Vol. 3. n° 2-1986.
- 1
},
such that ...,
we can
h)(z))
write
146
P. BONCKAERT
For the
right hand side we get
Intermezzo:
If E is a normed space, if L : E ~ E is an invertible SUBLEMMA. E is a continuous i-linear map then continuous linear map and if B : -
I
where B
(L,
...,
L)
I -r.. "
-
I -
T ’I. I 1
/T
is tne i-linear
map
1
1 T - ~1 II 7
aennea
oy
~’roof : 2014 Ubvious./. If
Let
we
us
apply this
make
an
in the
equality
estimate tor each term or tactor
separately
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
As
we
already
saw
in lemma
(2 . 3 . 5) :
symbol, as well as
Remark that this 0
R3
147
,
the
following,
ones
are
independent
1») because of the induction hypothesis ; this 0-symbol is Dih(z), by a independent of h since we can bound the II h E of constant independent F;’. =
...,
2. + 1k = In each term ot this summation we have j1 + So there must be 1, 2, ... , k ~ with jp f - 1. Hence, except for the case that ji1 jk = 1 and k = i, each term contains a j2 1 ~)_ factor Dkh(z) with k E {0,1, ...,f - 1 }; such a term is Suppose that ji = j2 = ... = jk 1 and k i. We can write ...
=
=
=
...
=
as a sum
of terms
containing Dh(z) as a factor, except the terms ot the torm
if
which is
=
we
replace (x, y) by h(z)
w e nave :
So if
write h =
(h 1, h2) ~ ( DF’Oh(z)) - h~i~(z) ~ ~_ ~ ~ we
Vol. 3, n° 2-1986.
we
get, since (x, y)
(1
+
~~x~ Y~
must be
replaced by h(z) :
148
P. BONCKAERT
e) Finally We end at : il1 i
because P
I
Q -
(2 . 3 . 7) LEMMA h(0) 0, if
1
0.
and a2
. - If h :
[0,5]] +
=
[0,5] Proof
on
D
00
-
B(o, ,u) is invariant under on ]0, ð] ] then h is
and if h is
if C x
0.
=
For all Z E ]0, ð]] we consider the sequence The invariance of the graph of h implies that we can write -
z. for alli E N, with in particular zo(z) z E ]0, zb] is of the form zi(z) then If Za E ]o, ~] and zb each zl(za) for some z E In order to prove the lemma it hence suffices to show the following: > 0 such that for all zo E [Zb, za ] for all j, s there exist za E ]0, 5 ] and and all i =
=
II
we can abbreviate this
shortly
zi
B B II ----
by writing
Let j,
=
From the
~~/1)/-/-
se
B1
>
=
Let us write
formula
recurrence
on
F~m we derive the existence of za
>
U
then for alli >_ 1 :
0 such that if
lor small za since a
/-/-
~J.
tor alli > 1 and from the estimates
and
v
u.
Applying this successively
we
get
~-11
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
[R~
149
On the other hand
II
," I
If
then for
Q - 1 ).
P Thus in that
(remember :
case II
1 /
B ) )
Choosing
we
have for
can find for all f =
We
constant
a
Ls such that for all
ze
[zb, za]:
~n(z)~ ~ Ls. ~~
t-1
From
+ t-i1
Vol. 3. n° 2-1986.
~2~ ~)
we
derive 2014~~~~ ~ "
so
So
150
P. BONCKAERT
We obtain :
for all i E Thus the case j 0 is done. Now by induction on j we prove that differentiate the equality =
for all
=
We
s e
j times (where h(z) _ (h(z), z)). Proceeding for that like in the proof of lemma (2 . 3 6) -
~
!!’B
. ~
_.
_._
we
find for the left hand side :
_..
__
where Ak is some ceo function; it is important to observe that Ak is linear with respect to the variable For the right hand side we get :
where the terms in the last summation +
oo we see I ir.
never
contain
Since
that
i_m _
’B I -
i
~
1I
AI - 0 - 1B
exactly like in the proof of lemma (2. 3. 6) which a priori might be unbounded ; which is, by the induction hypothesis, an
For the other terms we proceed but collect the terms containing
those terms contain a factor for any N E fil. Also -
So
we
-
_
n
v
.
T’Bo
v
_i v
v
find that tor all
Choose N
=
Q
+ s ; then we can find constants al
0 and B
j
>
0 such that
Annales de l’Institut Henri Poincaré - Analyse
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!R
VECTOR FIELDS ON
This estimate is of identical type the case j = 0. Now we may go obtain the desired result. D
as
the
on
151
started from when solving in the same way in order to
one we
exactly
(2 . 3 . 8). LEMMA. - There exists a neighbourhood V of (0,0,0) such that all orbits in V tend to (0,0,0) and those outside 1R2 x { 0 } are graphs of maps h satisfying the hypothesis of lemma (2. 3 . 7).
Proof - By means of the obtained Coo invariant graph f (h(z), z)z E [0, ð] ] ~ define the Ceo coordinate change
we
....
Put X G*X and write X riant we observe that =
From the
-
Since
Xy,
=
_
X
leaves the
inva-
expression of X we calculate that, for the standard inner product
[R2:
on
-
Because G does not alter -~
~ "-
......
oo
-
-
jets in (0, 0, 0) we can write for X : -"’"
_n
~.-
-.
-.
~
.
~
.20141
b 0 Let us write again (x, y, z) instead of (x, y, z). Since /(0, 0, 0) a and constant a we V of and since a can find 0, (0, 0, 0) neighbourhood c > 0 such that on V : =
~
^-
From this the result is Lemmas used in the
(2. 3 . 9) that
LEMMA. c
~......-
...... _
easily derived.
D
proof of proposition (2 . 3 ..1 ),
-
For each
>
case
II.
1, there exists
an 8 >
F§l’.
Proof The differences with the proof of lemma (2 . 3 . 6)
for
b) :
Vol. 3. n° 2-1986.
are :
0 such
152
P. BONCKAERT
for
e) :
LEMMA. There exist ~,~ > 0 such that if x in (0, 0,0) and j~, zi) B(0,~) [0, £ ] with lim sequence
(2.3.10)
is
-
a
=
.
graph of CQ coordinate change
then this sequence must lie
Proof -
we
With the
obtain for F
=
on
(Fx, Fy, FZ)
the
:=
G*F
that
0 on some neighbourhood of(0,0,0). for some a 2 Denote the transformed sequence (Xi, yi, Zi)iEN. We haveI ( xi + 1, Yi + ~ ) II if 8 is small. So then for all i : 1 1 1 .. ~ _ IIl 1 ~ 0 form -~ oo. Thus Xi Yf= 0, that is: the sequence lies on the z-axis, which is the transformed of the graph of hQ by G. D
( x_i,
=
(2 . 3 .11 ) LEMMA. - There exists a neighbourhood V of (0,0,0) on which all orbits of X starting outside the invariant graph leave V for t
-
-
Proof and
on
oo.
Precisely like in lemma (2.3.8) some neighbourhood V: -
~ rom this s a
we
obtain for
some
B
that the function V : ~3 -+ [R: (x, y, z) -~ ~(x, > 0, whence the result.
we see
Lyapunov function for X in the region z
emma used in the
proof of proposition (2 . 3 .1 ),
case
>
0
y) ~ ~ 2 D
III.
the map (Y, Z) : (2 . 3 .12) LEMMA. - For small /1,8> 0 and h E R2 : (y, z) F2(h( y, z), y, z) is a diffeomorphism onto [0,8]] - , ] at least) [2014~,~]] x [0, 8]. -
-
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F’2~~ Y~ Z)~ Y~ z) _ « 1 +
z)~ Y~
Y~
where Roo is 00 flat along the So, in a short notation : ~
-
We have:
153
+
Y~
and has
z-plane
T.~
..,.
)R~
~ _...
Z)~ Y~ z)) +
zero
~--
Y~
z)~ Y~ Z)
z-component.
D1
D"",
D ~ ........
W
D(Y,Z)(0,0) Identity. Because of the inverse function theoand because for all h E > 0 such 1 there exist [[D h(y, that for all h E the map (Y, Z) is a diffeomorphism on [2014~,~] x [0, a]. Put Roc 0). If 8 is small enough we have for all y E [- ~c, /1] : =
rem
=
C
and tor all ze ~it
As
there exists
Proof -
..
-- B
~B
~~ P
.. n
x
is
simply connected,
LEMMA. - There exists a J1 an 8 >
We
0 such that
=
rh. For i )
ims
t
impnes
1
> U such that tor all
c
(2. 3 .12) holds. and ( y, z) ( y, z)(Y, Z) we
such that lemma
choose ~
Let h E F~, H
T
Suppose, by induction
lUl
m
~U, 8j:
...Pwi~ I ~ .
(Y,z~[2014~~jJ
(2 . 3 .13)
~~
17
=
0
r’""7"B. II
on
1, ... ~2014 1- J
=
II
B
n7 i
)).
have:
AI nDB
i, that
1
tlllU
=2014
1.
A ’ LJ1
brevity put
and put also Vol. 3, n° 2-1986.
Fi1
=
Fx, f’2 =
It
we
differentiate the
equality
154
P. BONCKAERT
H
i times then
F2 0 h (2.3.6) that
=
We estimate
as
we
obtain in the
same
way
in lemma
as
follows :
h)( y, z) expression for a) From we see that we can write (2.3.12) the
=
D(Y, Z)( y, z) obtained
lemma
where Mh( y, z) is some matrix with M~(0,0) 0 ; this because Q see the rescaling construction. Moreover we can choose > 0 such that for all m E and all ( y, z) E [ - ,u, ,u ] x [0, e]: =
-1 > P > 2 : m
>
1, all
Then
because of the induction on the llAk(h( y> Z» > y> ...,
hypothesis z) II.
and because of the uniform bounds
1
because here the same here in particular
reasoning
as
in lemma
(2. 3. 6) applies and because
for
each q E ~0, ...,~ } d ) From tnnales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
we
1R3
155
obtain
e) r many :
ii s is small we nave ior an t
E l u, ... , m j :
~~a .I
I
n/-11; -
Hence
(2 . 3 .14) LEMMA. - If E is small and if (xt, y~, [2014~,/~]] x [- /1, /1]] x [0,8]] with lim (~,Zf) (0, 0) =
v
T2014’/
is and
a
sequence in
B/’~~B v
/
Proof. Similar to the proof of lemma (2.3.10) with this time the coorchange -
dinate
r
Lemmas used in the
(2 . 3 .15)
is
a
~- 20142014 TT 20142014 ~~Yl7 7~
proof of proposition (2.3.1),
LEMMA . - For
sufficiently small 8
diffeomorphism onto at least [2014 ~ /1]] We have (X, short notation :
Proof so, in
a
1 for
As [[
independent
of h.
Vol. 3, n° 2-1986.
z)(x,
z +
x
>
case
IV.
0 and for h
E
the map
[0, 8].
h(x, z), z)) + R ~(x,
remark that the last matrix is
z), z)
an
156
P. BONCKAERT
ea
D(X, Z)(0,0) =
We have:
0
.
Because of the
remark
foregoing
> 0 independent and the inverse function theorem there exist small is a diffeomorphism. such that (X, Z)[ on For all x E [- /1, /1]: 8 + EQg(x, h(x, 8),8) > 8 if ~ is small because of the estimates on g on V. + Also for all ze [0, s] and 8 small h(,u, z), z) > /1 and X
~(- /1)
h( - ~c, z), z)
+
-
,u.
Hence, since (X, Z)( [ - ,u, ~c ] x [0,6]) is simply connected,
LEMMA.
(2.3.16) m >
1, there exists
Proof
a)
We
an
Copying
-
We put F
-
=
can
There exists a J1 8 > 0 such that
0 such that for each
>
c
the scheme of lemma
(Fx, Fy, F )~ Fi = Fy~ FZ
=
(2. 3. 6) this
time
one
has :
F~).
write :
hence
b) and c) : d ) here :
the
same
so
for small
z
since
0 and
bi
>
0.
D
Annales de l’Institut Henri Poincaré -
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157
VECTOR FIELDS
~~ . ~ .1 i ~J
LEMMA.
-
11 8 is Small ana 11
lim
yi,
IS a
sequence
ill
and
(Y~~ z,) (0, 0) F(x;, y~~ z,)=(~-i, y~~ Z-~) [0, s] with [ - p, /1]] > the then must lie on this (i 1) sequence graph of hQ. the to of lemma 0 proof Proof - Analogous (2.3.10). x
(2. 3 .18)
LEMMA.
-
which all orbits of X t
-
=
There exists a neighbourhood V of (0,0,0) on outside the invariant surface leave V for
starting
DC.
-
Proof
- After the coordinate
change
r -
-
obtain a vector field X GX whose y and form (write again x, y, z instead of x, y, z) : we
:=
,
D
.
anu
wnerc j
Take a bounded such that on V :
are
of the
~.. ,
neighbourhood ~l
V
of(0,0,0) and constants a2, b 1, b2 > 0 -
Now the lemma easily follows. D This completes the proof of proposition ALL
components
u.
v
(2.4)
z
(2.3. 1).
THE EIGENVALUES ARE ZERO.
Here we will meet all situations II. A, II. B, II. C of the main theorem. Since we work with germs X of flatness zero, that is: 0, and since the z-axis must be formally invariant under X, we can assume, up to a linear change of coordinates preserving [R2 x {0} and {0}2 x [R, that the
1-jet
lines of this
is y~ ;equivalently: the matrix A introduced in the first c.~
chapter
IV is:
fo ana
Co
=
u.
Vol. 3. n° 2-1986.
it
158
P. BONCKAERT
of the previous and (2 . 3 .1 ). With (2 .1.1 ), (2 .1. 3), (2 . 2 .1 ) « reducing » we mean : to deform the vector field by means of (sometimes degenerate) coordinate changes which do not alter the nature of results such as for example « having oo contact », « being a cone of finite contact », etc. In fact all the cited cases will occur. We will try to reduce, in in propositions ( 1.1 ),
sense, this
some
case
to one
cases
(2 . 4 .1 ) PROPOSITION. Suppose X E G3 satisfies i ) the z-axis { 0 } 2 x [R is formally invariant under X -
ii) X is non-flat
Üi)
along
the z-axis
then
[R2 whose graph (germ) a) there exists a Ceo germ h:([0,oo[,0) z oc E is invariant under X and with j~(0) 0 { (h(z), z) [0, [} a exists cone K of finite contact there b) around { 0}2 x [0, oo [ such that -
=
in K
we
have situation II. A, II. B
or
II. C of the main theorem.
lemma (11.3.2.4) of X is of the form
Proof - Applying nent
0 0, 0 0 with y(0, 0)
where
is
fi, f 2,
~
0. Since j1 X
gi, g2,
hi, h2,
(0) == y~y 2014 ~x
y and
we
we can
S~
are
vi ) S, is a germ of a vector field with oo flat along [R2 x { 0 }. Let
me
explain
and y ; property
this
a
may
little bit. First
we
assume
that the z-compo-
in the ffollowing 11 win form : write X m
Cx germs and
zero 2014 -component and which az collected all terms linear in x that the z-axis is formally
ii ) together with vi ) reflects
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
invariant; property iii) simply means that hl and h2 don’t contain terms linear in x and y; property vi) indicates that we « absorbed » the ~ terms of the component of X in y. cz Allthough it is not crucial, we can spare some ink and calculations if we observe that we may assume 0. Because if not, we replace X by .
-
this germ is Ceo equivalent with X put the matrix
by means of the identity
map. We want
to
r~/-B
i i
handy form by means of a coordinate change. fl + g2 denote the trace of this matrix. This important role in the sequel. in
a more
Let T
If
we
=
play
an
put r
then
trace will
one
n 1
1
easily checks that
Note, by the way, the characteristic
that 1 (f1 - g2)2 4
equation
+ g1
is 1 4
times the discriminant of
of M. All this suggests the
following coordinate
change : /
As
Vol. 3, n° 2-1986.
1
~
’
160
P. BONCKAERT
straight forward calculation that the new vector in a point (x, y’, z) = ~(x, y, z) can be written in the following from : obtain from
we
a
where hl, h2, y, S ~ satisfy analogous properties as ii), For brevity we write
we can
write down
iii), iv) and vi) above.
~
~
Then
~,~X
in the form
which will become clear in
(new h2) :
perform
For
reasons
z
u~ of the z-axis, just like in the proof of proposition (2. 3. 1)
=
by
means
we always restrict our Calculating straightforward we find
us
a
moment we
a
rescaling case
III,
of the map
Remember that
Let
field
attention to the upper halfspace. if and only if that R*X’ =
simplify the notations by writing again X, y, z,
h2, Q, y, S x instead
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS 0~
of respectively X’,y’, u, h1o R,
put v(u)
=
T(u2~.
Then _
we
h2 R, 2Q - 1, o
something
obtain ,
161
!R 3
1 2 o R, S x
R ; let
a
us
also
of the form: ,
~.
_
where, for the sake of clearness, the following properties hold : /1""’Io.’B.
..... ,.. ~ -...
’"
..
v) Q > 2 a and which is vi) S~ is a vector field with zero --component az along ~2 x {0 }. Let me explain why it is no restriction to assume that
oo
flat
perform a blowing up idea similar to the one in the proof 01ofprothe position (1.1) as follows. Blowing up X n times gives a vector field
We
can
form r /i
1 ~
Sx satisfying similar properties as ii ), iii ) and ri) above. 0, The fact that y(0, 0, 0) # 0 should explain our assumption exist there ..., Q 1} which we take for granted from now on. So
for
h2,
some
and
such that
Vol. 3, n° 2-1986.
162
P. BONCKAERT
~
The to
«
«
strongest »
term in the
~
~
~
expression of X
is
of
y- . We want
course
C/~L weaken » it with respect to the other entries in the matrix
This situation is comparable with the problem : diminish the entries « 1 » emerging in the Jordan normal form of a square matrix. Inspiring ourselves on this, we consider for each P~N the coordinate change
P will be chosen in
is
a
we
For
for
sort
of partial
a
according
moment,
to the
occuring
cases.
In fact ap
blowing up.
have
our
vector
some
field here this
gives :
satisfying property u~)
above.
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VECTOR FIELDS ON
!R~
163
Intermezzo.
Let
me
indicate here
why it was necessary to rescale the z-axis. Take the
example : rr.l
Since the only allowed choices for P are here P 0 and P 1, we cannot weaken the entry « 1 » without creating a similar problem. However after the rescaling we have =
rA
and
by taking P
=
1
we are
End of the intermezzo. We distinguish the cases Case jP1g(0)
Choose P
Here
Since
we
=
lead to the matrix
=
Pi. We distinguish Pi
Q have
=
Vol. 3, n° 2-1986.
11
0.
0 and
0.
=
Subcase Pi
=
0
Q -
1 and
Pi = Q -
1.
(write again x, y, z instead of x, y’, z)
we
may write that
1.
164
P. BONCKAERT
by zP. We can write :
We devide the vector field 03B1P*X 2014 /,
_
2014t ~
B
1
In order to get rid of the possible unboundedness of the terms P hl(x, yzP, z) z 1 and 2P h2(x, yzP, z) it suffices to blow up the latter vector field 2P times z
(without dividing of course); indeed: thanks
to the
property iii) above
we
have ,
if we blow up 2P
P...
,
...11")...
í"B./II
times, the formulas in (II . 3 . 2. 3) imply that
factor z2P. This blowing up construction does not alter the This 1-jet is :
there appears
a
B
i,
,
1
1-jet
in 0
of ~
~
.
Since a 1 # 0, we clearly have a vector field satisfying the assumptions of proposition (2.1.3). The conclusions of proposition (2.1.3) remain valid for our original vector field because of the following reasons :
a) 03B1-1P 1( { (h(z), z) z E ]0, of a
b)
map
oo
if we have
tangent
a cone
oo
[ } ) u { (0,0,0) }
to the
z-axis in 0,
as
is the germ of the well as
graph
K of the form
then 03B1-1P 1 transforms it into r
I
This set contains the __
.
,
.
cone _
_1
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rn wnicn we nave
or
165
ui the mam theorem
the case; finally the rescaling R transforms this last
accoramg
to
f/.~ .. -B2014m)2 ..
Subcase With
field
P1 = Q a same
rn - -
cone
rI
7
into
.
7
IB. B. ’")
/
D ,
1.
construction
z ( ~ Y~ )
as
in the subcase above
with this time
we
2014component p oz
a
obtain
a
After
’Y( ~ Y
up 2P times (in order to get rid of unbounded terms) field satisfying the assumptions of proposition (1.1). We can conclude just like in the subcase above.
blowing
vector
we
obtain
a vector
Case jP1
g(o) ~
There exist
0.
P2 E ~ 1,
and ~2 ~ 0 such that
...,
_i _2~
Choose P
=
-
~2Po .
I
~/_2Po+2B
P2 . We distinguish three subcases : P2
Pi1
Q - 1,
or
Subcase P2
P1 ~ Q -
1.
We have B
w e aeviae inis vector neia
Vol.
3,
we can
n° 2-1986.
oy
i ~
1
z- ana obtain :
c
1
Agam
.
get
nd ot the unboundedness ot some terms
by blowing up
166
2P
P. BONCKAERT
times, just like above.
Y. The
Denote the result of this
blowing up construction
by
1-jet
of Y is
’"’
-,
0, then the eigenvalues
If a2
I and 0. Hence
are
apply proposition (2 . 2 .1) to obtain the situations II. A or II. B 0» of the main theorem. The same remarks a) and b) in « case jplg(O) hold. If a2 > 0, then the eigenvalues are 0. So up to a linear Y satisfies change of coordinates preserving ~2 x { 0 } and {0}2 the assumptions of proposition (2 .1.1 ). Blowing down this situation, we still have a Cx cone K of finite contact and a 2-dimensional C° subcone S like in situation II. C of the main theorem. Next contains 4 C x 1 cone K’ of finite contact just like in the foregoing case; observe also that 03B1-1P ag preserves the property «having 00 contact with the z-axis » ; S’ is a C° 2-dimensional subcone of K’ ; the rescaling R also preserves the property « having oc contact with the z-axis in 0 » ; R(K’) contains a C’~ cone of finite contact and R(S’) is a C° 2-dimensional subcone of R(K’). All this shows that, for our original vector field, we are in situation II. C of the main theorem.
we can
=
a2 , - ~/~2,
:=
Subcase
P2
We have
Dividing
=
Pi
Q ~
1.
,
this vector field
,
,
by zP r
i
we
obtain : ,,--,
-
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mow tms
vector
up
zr times to
field has
a
get
ON R3
167
na 01 ine unoounaea terms,
i ne
resuitmg
1-jet /1
/
v-_,
1
B
~
Since the trace of the matrix
is
a1(~ u)
according Next
we nave ai least one nonzero
we can
Subcase
P2
With
eigenvalue.
rience we can
to the case,
=
propositions (2 .1.1 ), (2 .1. 3), (2 . 2 .1 ) conclude just like in the previous subcase.
(2 . 3 .1 ).
vector field
1z (Xp*X
P1 = Q - 1.
a same
construction
as
above
we
obtain
a
~ with So
a
apply,
or
P
2014- component zy(x, yzP, z).
we can
apply proposition (1.1).
§ 3 . Proof
D
of the main theorem (1.2.1) with of the C° result.
exception
This is merely a summary of all the foregoing. Take a maximal directed sequence of blowing ups of X along D (definition III.1.3). If this sequence is finite, then apply proposition (III.1.4) to obtain situation I. If this sequence is infinite, then, using proposition (III.1.5), D is formally invariant under X. Theorem (111.2.1) implies that, after a finite number of blowing ups, we are led to a vector field (germ) of flatness zero. Now the theorem follows from propositions (1.1), (2.1.1), (2 .1. 3), (2 . 2 .1 ), (2 . 3 .1 ) and (2 . 4 .1 ), since these contain all the possible cases of flatness zero and since we assume that X is non-flat along D. D V. PROOF OF THE C° RESULT IN THE MAIN THEOREM (1.2.1)
§ 1. Definitions
and notations.
We want to provide « universal models » for the situations II. A, II. B and I I . C obtained in the main theorem (1.2.1). For that purpose we introduce : Vol. 3. n° 2-1986.
168
P. BONCKAERT
( 1.1 )
DEFINITION.
-
The
following
germs of vector fields
called the standard models for situation II. A resp. II. B resp. II. C of the main theorem (I . 2 .1 ). are
( 1. 2)
DEFINITION.
-
is called the standard
The germ of the set
cone
around the z-axis.
§ 2. Let us state here in a more the main theorem (I.2.1).
The C° result.
precise
way the assertation announced in
Let X E G3, let D be a direction such that X (21) PROPOSITION. leaves D formally invariant and such that X is non-flat along D. Then there exists a cone K of finite contact around D, a C° cone K’ K~ containing K such that the germ XK’ is C° equivalent with either or SBKS or Sc Ks, according to the fact that X is in situation II . A resp. II. B resp. II. C of the main theorem (1 . 2 . I). Moreover in situation II. B we have C° conjugacy. -
Proof this).
We may
assume
that D is the z-axis
(see part
III for comments
on
know that, after a finite number of blowing ups, after possibly rescaling of the z-axis, after possibly a partial blowing up and after putting in normal form, we are always lead to a vector field Y of one of the following types : (change the sign of X if necessary; with « flat terms » 0 plane) we mean terms Jo flat along the z From part IV
we
a
=
1) the eigenvalue of Y along the z-axis is to
the
z
=
0
plane
is
a
0 and the restriction of Y
hyperbolic expansion; Annales de l’Institut Henri Poincaré -
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169
VECTOR FIELDS
with
a>0, b>0, /i(0,0,0) ==~(0,0,0) 0, y(0,0,0)
0;
=
3) the restriction of Y to the z and the z-component of Y is 4) the restriction of Y to the z and the z-component of Y is
=
f(O,O)
=
6)
Y has the
7))
Y
0, g(0, 0, 0) same
3~
form
y, z)y 2014
10)
Y
=
same
y,
form
0;
in 5 but with this time
.f ( ~ Y~
+~
/(0,0,0)>0~(0,0,0)0; 8) Y has the same form 7(0,0,0)0; 9) Y has the 1’(0, 0, 0) 0 ;
0, y(0, 0, 0)
>
as
0
~B
/ ~
with
plane is a hyperbolic expansion z) with y(0, 0, 0) 0, Q > 2 ; 0 plane is a hyperbolic contraction z) with y(0, 0, 0) 0, Q > 2 ;
=
+
y( ~
g(0, 0, 0)
y, z) 2014 + flat +
0;
terms with a >
as
in 7 with this time a
0, f (0, 0, 0)
as
in 7 with this time a
0, f(O, 0, 0)
z)x ~ ~x
+
+
y,z) ~ ~z
0, 0,
>
0,
+ flat terms with
0, a > 0, y(0, 0, 0) 0 (nota bene : we have interchanged the and y compared with the situation in proposition (IV. 2 . 3 .1 )). One has the following table :
j(O, 0, 0) role of
x
Note that the operations (blowing up, rescaling, etc.) are homeomorphisms of [R2 x ]0, oc [ with the property that a sequence tending to the = - 0 plane is transformed into a sequence tending to the z 0 plane. =
Vol. 3, n° 2-1986.
170
P. BONCKAERT
Roughly spoken, the idea is to construct in the origin a local C° equivalence with the corresponding standard model blown up once. Moreover we take care that the homeomorphism, realizing the C° equivalence, transforms sequences tending to the z 0 plane into sequences tending to the z 0 plane. When returning back the whole way this will assure the continuity in (0, 0, 0) of the desired C° equivalence for our original vector field X. We assume that for each considered situation there has been blown up at least once. =
=
Situation II . A. We blow up
This
blowing
once
the standard model SA and get
up transforms the standard
cylinder { (x, y, z) x2
+
y2 - 1, z
>
cone
KsB{ 0 }
into the
(full)
0 }.
out just like in lemmas (IV. 2. 3. 8), (IV. 2 . 3 .10) we can that the z-axis is invariant under Y. We can find a small cylinder B(0, ~c) x ]0, 5 ] around the z-axis such that { (x, y)I X2 + y2 - ,u2 ~ x ]0, 5]] is transversal to the orbits of Y: for types 1 and 3 this is clear from the hyperbolicity (provided decent coordinates are chosen) ; for types 5 and 7 we’ consider the function G(x, y, z) = x2 + y2 and observe that for type 5:
- By flattening
assume
:=
y, z) + flat terms and for type 7 : ( VG(x, y, z), Y(x, y, z) ~ ==2(x~ + 2ax2 + + flat terms ; VG(x, y, z), Y(x, y, z) > == f(.r, ~.7, in both cases we have VG(x, y, z), Y(x, y, z) > > 0 on V/l-,ð provided ~ and 6 are small; hence the orbits of Y are transversal to the level surfaces of G in
Moreover, if 6 is the planes z zo, 0 between Y |V ,03B4 and chosen such that =
To
see
this
we
small, then the orbits in
are also transversal to 03B4. Now it is easy to construct a C° equivalence SA |{x2+y2~ 1 and 0z~03B4}; the homeomorphism h can be
proceed
as
follows. Take the
homeomorphism
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
Extend n to
171
as follows: u
then its
positive
orbit for Y intersects {(x,y, z) |x2 + y2 /12, 0 z 5} a first time in a point, say, define y 1, yo, zo) to be the intersection of the of orbit the plane z for with negative SA z1) Zo; finally define for 0 ð. 0, 20) = (0, 0, zo) zo =
N
=
Situation 77. B. We blow up the standard model
SB and get ~
N
1 nis vector neia is transversal to tne 1-1 _ 7 ~v
1x .
,1
.,
I v-2
-
nanspnere.
-v-2 _J ,,2 ,I
,,2
....L
transversal 10 me orons 01 Y : ior
....L
type
-2 _ A
no...
’7~ 2014 ~~ ~~, 4 tms ciear from the
hyperbolicity
and from y(0,0,0) 0 (provided decent coordinates are types 6 and 8 we consider the function G(x, y, z) = x2 + observe that for type 6:
m both
~BB
..
~,
_
1_
~t100
/~~ /
Consider the
homeomorphism .
--
--
I
y2
’)....0+ 1...~~....
+
z2 and
_B
U on a small set
cases ( VCJ(x, y, z), Y(x, y, z) > ~~
,.,.B
w
chosen) ; for
.. .
~~
...
2
, "’B
..
Extend h to Vð as follows. Let and 03C6B denote the flows of Y resp. SB’ For each (x, y, z) E Va there exists a t >_ 0 such that ~y(2014~ (x, y, z)) ~ Hi. We force the conjugacy to be true on Va by defining LI - -
we cnecK me
-
-B B
JL
required property
/.
1_l 1 /~l__ __ _B111 B
ior
yi,
oe a
sequence
m
v03B4
denote the sequence 0). Let (ti times » such that ~(2014f~ Suppose by contradiction y~ the z-components of a subsequence would stay
tending to the z of « that
-
Vol. 3, n° 2-1986.
=
0 plane (that is :
lim zl
=
172
I
P. BONCKAERT
M
>
0 for
some
constant M. This
implies
that sup tik
+
Hence
~6~
ie
z-components of ~y(2014~ (x~k, ylk,
o we
have
a
decent
conjugacy
h between
tend to zero. A contradiction. and
preimage h 1( ~ (x, y, z) + y2 1 and z > 0}) still contains a ylinder around the z-axis in Va. Returning this situation the whole way ~ack to our original vector field X this gives us the desired cones and onjugacy. .’he
ituation II . C. We blow up the standard model
Sc and get
1 the level surfaces of g are transversal to the Hence in the region z of orbits Sc. Take 0 b~ 1. Take a2 > 0 so small that the intersections of the level surface G-1( - ~2) with the planes y a2 and y a2 are circles C2 z the 1. See inside 2 from on. now figure region resp. D2 =
FIG. 2.
-
=
-
Sketch of the first octant. Annales de l’Institut Henri Poincaré -
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173
VECTOR FIELDS
We also consider the level surface G ~(1); this is a two-leaved hyperboloid. Next we consider the surface obtained by letting flow all the points of the circle C2 until they hit the level surface G - l( 1 ). We do the same x ]0, x [ a region R2. thing for the circle D2. In this way we enclosed in we that the out assume 0 By flattening may y plane is invariant under Y. We want to make an analogous construction as above for Y. We have for type 2: =
and for type 9:
ana tor
type
1U:
/ K7/U/,,
..
_B
So for all three types we can find in V:= W n ([R2 x ]0, t [ ) :
a
neighbourhood W
of (0,
0, 0)
such that
miniature version of the construction for Sc. 0 we can make the following construction inside V. Consider the level surface G - 1( ð 1)’ The intersections of it with the planes
we
want to
make
For small
5i
a
>
-
~ v
=
5i
and y = -03B42 03B12
surface obtained
by letting
hit the level surface
are
circles
flow all the
G-1(03B41 03B42).
Ci
resp.
points
We do the
Di. Consider also the
of the circle C i until
same
r
Second
we
Vol. 3. n° 2-1986.
extend h to
for the circle Di.
thing
In this way we enclosed a region Ri in V. Now we try to make an equivalence between define the homeomorphism
and ~
ç:
Ras follows. Require that
a
they
Sc
First
we
Î
level surface 7
174
P. BONCKAERT
(a1 e 2014 5i, 2014 )
is
mapped
onto the level
(x, y, z) E Ri lying in a level surface where the negative integral curve of time the level surface
of
G-1 (03B42 03B41 ai
we
surface
,
for
y 1, Z1)
consider the point
through (x, y, z) hits for the first we define G"~(2014 5i); h(x, y, z) to be the intersection
with the
Y
yB zl .
positive integral curve of Sc through
We check the required property for h. 0. sequence in Ri with lim zi
Suppose
that
(xi,
is
yi,
a
=
Let (xI
, y1i, z1i) be the point where the negative integral curve of Y through
hits for the first time the level surface G’~(2014 tends to the y 0 plane, we get that sequence (xl, yl, 0 plane. tends to the z D
~i).
Since the
=
yi,
=
(2 . 2) [Cam ] .
REMARK. - We have used
VI. SOME
some
ideas
comparable
to
those in
EXAMPLES, COUNTEREXAMPLES QUESTIONS
AND SOME
§
1. A
counterexample
and
some
questions.
might pose the question whether every germ in 0 E [R3 of a Cx satisfying a ojasiewicz inequality (see definition I.1.17) one-dimensional invariant manifold, or equivalently, possesses a whether it possesses an integral curve tending to 0 (in positive or negative time) in a Ceo way (by tending to 0 in a Cr way we mean : if we add the origin to the integral curve, we obtain a C~ invariant manifold). The answer is no. We give an example of a germ for which no integral curve can tend to zero in a C2 way; the germ has nonzero 1-jet. One
vector field
(1.1)) fies
EXAMPLE.
-
Let X
=
+
+
ojasiewicz inequality. No orbit of X Proof - a) Blow up X in the x-direction: a
x2 ~ ~z. can
Observe that X satis-
tend to 0 in
a
C2 way.
Observe that
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VECTOR FIELDS ON
has
just
one
singularity
in
(0, 0).
!4°
175
Blow up X in the x-direction : ,
-,
,
Observe that
nas no
singularities. up X in the
h) Blow
y-direction : ~
Observe that
has 11U
sing
c) Blow
up X in the z-direction : ,
Observe that
~~~-~
Now
we are
by blowing
able to describe the vector field X on S2 spherically (see for example II. § 1 or [Ta,
up X
obtained Du2] for defi-
nition and construction). The only singularities of X IS2 x {0} are (1, 0, 0) and ( - 1,0,0). If an orbit 811 of X tends to 0 in a C2 way, then X must have an orbit 82 which tends in 1 a C~ way to ( 1, 0, 0) or ( -1, 0, 0) since these are the only singularities on S2 x { 0 }. If we blow up X in (1, 0, 0) or ( - 1,0,0) we don’t have a singularities any more, except in the « corner » : see figure 3. This contradicts the fact that 82 tends C1 to (1, 0, 0) or ( -1, 0, 0). Q
( 1. 2) REMARK. - The vector field X in example ( 1.1 ) must have an orbit in the first octant(.v, y, ~) E [R3I x >_ 0, y >- 0, z > 0 } tending to 0 in a C° way. One can see this as follows. Denote Vl = {(x, y, z) E [R3 (x 0 or y 0 or z 0) and x > 0 and and and and z > 0 r > 0 x+y+z 1} V2 == { (x, y, z) e ~ ! jc > 0 =
and y > 0 and Vol. 3, n° 2-1986.
z >
=
=
0 }. V3 is the first octant. See figure 4.
176
P. BONCKAERT
FIG. 3.
-
Blow up X in
(1,0, 0).
FIG. 4.
Each t
-
point of vl B ~ 0} +00. This follows
the function
G(x,
y,
enters the first octant
from the Since
immediately
z) x + y + z. VG(,r, y, z), X(x, y, z) ) = =
V3 and
y +
z
leaves it for of X. Consider
never
expression + x2
Annales de l’Institut Henri Poincaré -
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177
VECTOR FIELDS
the
positive
consider the
orbit of
a
point
.-........-
""""’4.""""’’’’’’------
T(u) _ -
V1 B { 0 } intersects V2 L,)f 0 } of a point
negative orbit {
£
If
of
‘,.~, .. ~ J
’JJ, then
T/,.B _
.
lim
necessarily
r) leaves V3 in negative time in
some
’B1 in t ~
a
point
or
V2
v E
to 0 if
to the
lim
u)
-
r E
-.---......
point,
---"’’’’’’’’’0
0. If
=
Conversely,
V~. Put
..~- 1 - n
I
-v-
L --, ..&. s...1..~,..__J
once.
----
T(v)
---------------
> - ex)
~_.
then
say,
r.- 1,; ,
point of Vi where the negative orbit through v hits Vi v) 0. Certainly ViB { 0 } since the positive =
orbit of each point of V1B{ 0 } hits V2. Since V2 is compact and since V1B{ 0 } is not compact, necessarily h(V2) V1. Hence there must be a point in V2 tending to 0 for t - - 00. =
I don’t know whether this vector field X possesses (1 . 3) QUESTIONS. orbit tending in a C~1 way to 0. I generalize this question : does there exist a germ in 0 E [R3 of a Cx vector field satisfying a xojasiewicz inequality without an orbit tending I to 0 in a C° way? in a C~ way? (in positive or negative time). -
an
vector fields having a Coo dimensional invariant manifold.
§ 2 . Examples of one
A very
general observation from
the
chapters
III and IV is :
OF THE PROOF OF THE MAIN THEOREM. If X E G3 D direction leaves formally invariant (that is: invariant by the oc jet) and if X is non-flat along D then there exists a C x one-dimensional invariant manifold x tangent to D. D If we consider germs in 0 E [R3 of vector fields satisfying a Mojasiewicz inequality, it hence suffices to look for conditions which guarentee the existence of a formally invariant direction. In general it may be a difficult task to investigate whether a given vector field has a formally invariant direction. Some tools for this can be : apply the normal form theorem (which we will do in some examples hereafter); or: try to blow up the vector field until you find a singularity having a formally invariant direction by the normal form theorem. An important result in this sense is :
(2.1) CONSEQUENCE a
(2.2) THEOREM [B. D. ]. - If X E G3 satisfies a ojasiewicz inequality DX(0) has eigenvalues 0, i i~. - i i ( i E (1~ B ~ 0 } ) then X has a C x
and if A
:=
3. n ~-1 y~b.
178
P. BONCKAERT
one-dimensional invariant manifold D tangent in 0 to the rotation axis of etA, t # 0. Moreover there exists a cone of finite contact around D in which we have situation II. A or II. B of the main theorem (1.2.1). In the same spirit we can apply the normal form theorem to other types of singularities with nonzero 1-jet. For many cases the result is well known, but let us list them for the sake of completeness :
(2 . 3)
MORE
EXAMPLES.
-
Let X e G, Xi = DX(0).
Annales de l’lnstitut Henri Poincaré -
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VECTOR FIELDS ON
Proof.
179
2014 Denote _
where
I~$3
A,
are
_
(~
.
real constants r~
_
~
(zero rv
is
1
allowed).
Put
i
where h > 2 (see formulation of normal form theorem also denote, for h >_ 2, 0 j h, 0 i _ j :
(I. 1 21 )).
Let
us
i ms is a oasis ior n .
After
a
trivial calculation
one T I
finds that B
where
So D has On Hh
a
diagonal matrix with respect to this basis. put the standard inner product with respect
we
Vol. 3. n° 2-1986.
to
this basis.
180
P. BONCKAERT
For as
O? c002 ~ 0. example 1 we see that complementary space (Im D)1; an element
2014
contain the terms e001 = zh
nor
3.B’
e002 = zh
e002 ~ Im D. Take
so
(Im D)1
of
2014
cannot
since
~y
Hence in the normal form the z-axis is formally invariant. For example 2 the same reasoning applies; for example 5 we see that ehn3 E Im D and we can follow an analoguous reasoning. A similar method works for examples 3, 4, 6. Next we calculate that
so
S 1 has a matrix of the form
where A is an upper triangle matrix with zero diagonal elements is nilpotent). Then one easily calculates that S 1 must be nilpotent. So for examples 7, 8, 9 we can write
(so A
where D is semi-simple and Si is nilpotent. The fact that Im D c Im [X 1 - ]~ implies, in the same way as above, the results claimed in examples 7, 8 and 9. Concerning example 10 we find that the matrix of S2 is of the form
where B is an upper has a matrix
triangle matrix r
and
one
easily
..
-
with vi
calculates that S3 must be
zero
diagonal
elements. So 83
/~ ~)
nilpotent.
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181
VECTOR FIELDS
[X 1, - ]~ D + S3 again reasoning as above can be made. For examples 11 and 12 we find As
=
,
so
e001, e002~ Im
[Xi, 2014 ]h
1m D
c
1m
[7(1, - Jh
and
the
same
that ’B
~
and
we can reason
like before.
D
In case j1X(0) 0 there is, as far as I know, very (? . 4) REMARK. little known on normal forms. But as already said, blowing up may sometimes help. This is the case for a C x gradient vector field X = grad f : F. Takens 0 then X has showed me, using a blowing up argument, that a ex invariant manifold, in all (finite) dimensions. In example (1.1) 0 was not an isolated zero of the first nonvanishing jet. Even the assumption that 0 is an isolated zero of the first nonvanishing jet (which implies that the radial eigenvalue in a singularity of X IS2 x {0} is nonzero) is not enough to have an invariant direction : =
-
EXAMPLE.
(2.5)
Observe that 0 is
-
an
Let
isolated
zero
of J2X(0). No orbit of X
can
tend to 0
in a C2 way
Proof - a)
Blow up X in the x-direction : a
i
i
nas
just
one singularity
m
(U, UJ.
Blow up Xx in the x-direction :
and observe that
has
no
singularities.
3. n° 2-1986.
~
,
182
?. BONCKAERT
b) Blow up X in the y-direction : and observe that
has
no
c)
singularities.
Blow up X in the z-direction : ~
nd observe that
has no singularities. Now we can conclude
just
like in
(2.6) .REMARK. This example This can be seen from the expression
example (1.1).
D
to 0 in a C~ way. is a hyperbolic expansion.
has orbits tending
(2. 7) SOME FINAL REMARKS. - a) Concerning the main theorem (1.2.1): when X is analytic, it is not necessary that (one of) the obtained invariant manifold(s) is also analytic, as was pointed out to me by the referee: for X
=
x - + ((yY - z2 a ~+ z"2014OZ
the z-axis have
an oo
all the invariant directions tangent to g
jet in 0 of the form
b) Concerning proposition (V . 2 .1) about C° equivalence with standard (but cannot prove) that « C° equivalence » can be « C° replaced by conjugacy ». models: I presume
REFERENCES [Ar] [A. M.] [A. R.] [B. D.]
[Be]
V. ARNOLD, Équations différentielles ordinaires, Éditions Mir, Moscou, 1974. R. ABRAHAM and J. E. MARSDEN, Foundations of Mechanics. Benjamin, London, R. ABRAHAM and J. ROBBIN, Transversal Mappings and Flows. Appendix by Kelley, Benjamin, New York, 1967. P. BONCKAERT and F. DUMORTIER, Smooth Invariant Curves for Germs of Vector Fields in R3 whose Linear Part Generates a Rotation, to appear in
Journal of Differential Equations. G. R. BELITSKII, Equivalence and Normal Forms of Germs of Smooth pings. Russian Math. Surveys, t. 33, 1978, p. 107-177. Annales de l’Institut Henri Poincaré -
Analyse
non
Maplinéaire
VECTOR FIELDS ON
~3
183
H. W. BROER, Formal Normal Form Theorems for Vector Fields and some Consequences in the Volume Preserving Case, in: Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Math., t. 898. 1981. [Cam] M. I. T. CAMACHO, A contribution to the Topological Classification of Homogeneous Vector Fields in R3 (preprint). [Car] J. CARR, Applications of Centre Manifold Theory, Springer Verlag, New York, 1981. [Di] J. DIEUDONNÉ, Foundations of Modern Analysis, Academic Press, New York, 1969. [Du1 ] F. DUMORTIER, Singularities of Vector Fields on the Plane. Journal of Diff. Eq., t. 23, 1977, p. 53-106. [Du2] F. DUMORTIER, Singularities of Vector Fields. Monografias de Matematica, t. 32, IMPA, Rio de Janeiro, 1978. [D. R. R.] F. DUMORTIER, P. R. RODRIGUES and R. ROUSSARIE, Germs of Diffemorphisms in the Plane. Lecture Notes in Math., t. 902, 1981. [Go] R. E. GOMORY, Trajectories tending to a Critical Point in 3-space. Annals of Math., t. 61, 1955, p. 140-153. [Gu] J. GUCKENHEIMER and P. HOLMES, Nonlinear oscillations, Dynamical Systems and Bifurcations, Springer-Verlag, New York, 1983. [Ha] P. HARTMAN, Ordinary Differential Equations. Wiley and Sons, New York. [H. P. S.] M. W. HIRSCH, C. C. PUGH and M. SHUB, Invariant Manifolds. Lecture Notes in Math., t. 583, 1977. [Ir] M. C. IRWIN, Smooth Dynamical Systems, Academic Press, New York, 1980. [Ke] A. KELLEY, The Stable, Center-Stable, Center, Center-Unstable, Unstable Manifolds. Journal of Diff. Eq., t. 3, 1967, p. 546-570. [M. M.]J. E. MARSDEN and M. McCRACKEN, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [Na] N. NARASIMHAM, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam. 1968. [N. S.] V. V. NEMYTSKH and V. V. STEPANOV, Qualitative theory of Differential Equations. Princeton, Princeton University Press, 1960. [P. M.] J. PALIS and W. DE MELO, Geometric Theory of Dynamical Systems, SpringerVerlag, New York, 1982. [Sh] D. S. SHAFER, Singularities of Gradient Vectorfields in R3. Journal of Diff. Eq., t. 47, 1983, p. 317-326. [Sm] D. SMART, Fixed point theorems. Cambridge University Press, 1974. [Ta] F. TAKENS, Singularities of Vector Fields. Publ. Math. IHES, t. 43, 1974, p. 47-100. [We] J. C. WELLS, Invariant Manifolds of Non-Linear Operators. Pacific Journal of Math., t. 62, 1976, p. 285-293. [Ya] S. YAMAMURO, Differential calculus in topological linear spaces. Lecture Notes in Math., t. 374, 1974.
[Br]
l
3.
n° 2-1986.
-
-
-
--.-r
-
...____
--
._j._-__
-
-
____~__
-_
--
-
/
Vol.
3,
n°
Poincare,
2, 1986,
p. 111-183.
mooth invariant
curves
of
singularities
of vector fields
on
R3
Patrick BONCKAERT Limburgs
Universitair
B-3610
centrum,
universnaire
campus,
Diepenbeek (Belgium)
We describe singularities of Coo vector fields, mainly ABSTRACT. in ~3, in the neighbourhood of a given « generalized » direction (this is : the image of a Coo germ y : ([R + , 0) - (~3, 0) with y’(o) ~ 0). It is a local study. One of the major results is : if X is a germ in 0 E f~ 3 of a C~° vector field and if X is not infinitely flat along a direction D then there exists a cone of finite contact around D in which four specific situations can occur. In three of these situations D is formally invariant under X (with formally we mean : up to the level of formal Taylor series) and there exists a Coo one-dimensional invariant manifold having infinite contact with D. In particular, we obtain that the existence of a formally invariant direction D always implies the existence of a « real life » invariant direction having infinite contact with D, provided that X is not infinitely flat along D. Using the blowing up method for singularities of vector fields we reduce a singularity always to either a « flow box » or to a singularity with nonzero I-jet and with a formally invariant direction. Finally we give topological models for the obtained situations. I would like to thank Freddy Dumortier for suggesting me the problem and for his valuable help. -
RESUME. dans [R3 au
Nous decrivons les singularites de champs de vecteurs Cx voisinage d’une « direction » donnee, c’est-a-dire de l’image
-
Liste de mots-cles : Vector Field.
Singularity. Invariant Manifold. Classification AMS : 34C05 Ordinary Differential Equations. Qualitative Theory. Location of integral curves, singular points, limit cycles. Analyse non linéaire 86/02.111 / 73 S 9,301 ç Gauthier-Villars
Annales de l’Institut Henri Poincaré -
Vol.
3. 0294-1449
2
?. BONCKAERT
(R3, 0),
avec 03B3’(0) ~ 0. C’est une étude locale. si particulier que X est le germe a l’origine d’un champ C" ;i X n’est pas plat dans une direction D, alors il existe un cone de contact ~rdre fini autour de D ou quatre situations distinctes peuvent se produire. ms trois d’entre elles, D est formellement invariant par X et il existe e courbe invariante C~ presentant un contact d’ordre infini avec D. utilisant la methode de blow-up pour les singularites de champs de ~teurs, nous reduisons toutes les singularites soit a un « flow box », t a une singularite au 1-jet non trivial possedant une direction forllement invariante. Enfin, nous fournissons des modeles topologiques ur les diverses situations obtenues.
03B3 : (R+, 0)
m
germe
>us
montrons en
-
,
Je remercie Freddy Dumortier pour m’avoir ur son aide.
I.
suggere
ce
probleme
et
INTRODUCTION, PRELIMINARIES AND STATEMENT OF THE MAIN RESULT
§ 1. Elementary definitions
and useful theorems.
Let X be a vector field on ~n. We (1.1) DEFINITION. s a singularity in p E f~n if X(p) 0. If we only investigate local properties of a singularity in p, ction to put p in 0, the origin. -
say that X
=
it is
no res-
DEFINITION. Two vector fields X and Y on tR" are called germuivalent in 0 if there is a neighbourhood U of 0 on which they coincide,
( 1. 2)
-
DEFINITION. The set of all vector fields on [?" which are germuivalent with X (in 0) is called the germ of X (in 0). In the same way we n define germs in 0 of functions, diffeomorphism,... The germ in 0 of a set A ci {?" is the germ in 0 of its characteristic func>n A germ will often be confused with a representative of it if this
( 1. 3)
-
without
danger.
NOTATION. lds X on ~n with
( 1. 4)
-
Gn denotes the set of all germs in 0 of ex vector 0.
X(0)
=
DEFINITION. Let X, uivalent if their derivatives in 0 up to, and X(0) D’Y(0), Vi, 0 i k.
( 1. 5)
-
X and Y are called k-jet order k are equal:
including,
=
Annales de l’Institut Henri Poincaré -
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113
The set of all Y E Gn which are k-jet equivalent It is important to with X E G" is called the k-jet of X and denoted observe that the k-th order Taylor approximation of X in 0 belongs to we often will make no distinction between these two objects. In 1 correspondence between k-jets and vector fields X fact there is a 1 0 and with polynomial component functions of degree k. on [Rn with X(0)
(1.6)
DEFINITION.
-
-
=
(1.7) NOTATION limit of the sets Ji for the
We
mappings nlk : J?
can
take the inverse -
-
jkX(0) (1
>
k).
DEFINITION. This inverse limit is denoted J; its elements called oo-jets; the oo-jet corresponding to is denoted Elements of J~ can be regarded as n-tupels of formal power series in n variables; also j~X(0) represents the Taylor series of X at 0. In the same way we define jets of functions, diffeomorphisms...
(1.8)
-
are
(1.9) and
DEFINITION. 1 X(0) ~ 0.
X E Gn is said to
-
haveflatness
k if
=
0
Y are said to be Cr conjuif for some (and hence for all) represen1) tatives X and Y of X resp. Y there are open neighbourhoods U and V of 0 in Rn and a C7 diffeomorphism 03C6 : U ~ V such that Vx E U :
( 1.10) DEFINITION. gate (r E N u { 00, cv },
-
X(x)
=
Let
X, Y E Gn. X and
r >
ifi( 03C6(x)).
We also write:
~~X
=
Y in this
case.
Let X, Y E Gn. X and Y are said to be Cr equi(1.11) DEFINITION. valent (r E N u { ~, 03C9}) if for some (and hence for all) representatives X -
and Y of X resp. Y there are open neighbourhoods U and V of 0 in fR" and a Cr diffeomorphism h : U ~ V which maps integral curves of X to integral curves of Y preserving the « sense » but not necessarily the para[0, t ]) c U, t > 0, then metrization ; more precisely : if p E U and and q,v there is some t’ > 0 such that [0, t’ ]) h( [0, t ])). denote the flow of X resp. Y). Time preserving Cr-equivalence is called Cr conjugacy and coincides with definition 1.10 for r > 1. With C° diffeomorphism we mean a homeomorphism. =
(1.12) THEOREM (Borel) [Di, Na]. - For all such that T = j~X(0). (Or: j x : Gn ~ J) is
T e J) there exists
surjective.).
an
X E Gn
0
in ofor shortlya direction, (1point.C’-difieomorphic 13) DEFINITION. image A generalized direction diffeomorphism is
a
0. The
Vol. 3. n° 2-1986.
of the germ in 0 E of a halfline with endis assumed to preserve 0.
14
P. BONCKAERT
( 1.14) DEFINITION. - A germ in 0 of a set K m [Rm+ 1 is called a C‘ l > k, if there exists a germ of a Cj function h : ~olid) cone of contact k, oo with (~0) jkh(o) 0, ~+1~(0) ~ 0 and a germ of a C" [0, [, 0) (Rm+ 1, 0) such that iffeomorphism 03C6: (!Rm + 1, 0) -
=
-
x K is called a cone of contact k around the direction D 4>( { The intersection of a C7 diffeomorphic image of a hyperplane, through D, with K is called a Crm-dimensional subcone of K (0
[0, 00 [) passing r 1).
=
If D is a direction in [?", then there exists a germ of a map y : ([0, oo [, 0) - (~", 0) with y’(0) # 0 such that the image of y is D. Conversely, if y : ( [0, oo [, 0) - (~ 0) is Coo and if y’(0) # 0, then the image of y is a direction.
(1.15)
PROPERTY.
-
Proof 2014 Easy.
D
(1. 16) DEFINITION. - For XEGn and D a direction we say that X is nonflat along D if for some (and hence for all) Coo germ }’ : ( [0, CfJ [0) with y’(0) # 0 and image D we have. y)(0) ~ 0. In the other case we say that X isflat along D. ~
We say that X E Gn satisfies a ojasiewicz (1.17) DEFINITION. quality if there exist k~N and C, 03B4 E ]0, 00 [ such that -
PROPERTY. - If X E Gn satisfies non-flat along every direction.
(1.18) s
( 1.19) CONVENTION. or
a
-
vector field X on ~m
( 1. 20) DEFINITION.
formally
-
a
Zojasiewicz inequality
then X
We write (x, z) for an element of [Rm x; we write X (Xx, Xz). =
We say that X E Gm + 1 leaves the
invariant if for all
ine-
(0)
=
z-axis {0 }m x R
O. This is the
same as
saying
does not contain pure z-terms. 1 x [0, oo D is formally invariant under X E Gm+ A direction D ~( { ~ * 1 X leaves the z-axis formally invariant. =
f
( 1. 21 ) THEOREM (Normal Form Theorem, Poincare and Dulac). - Let (E Gn and let Xi be the linear vector field on [?" such H~ denote the vector space of those vector fields on !?" Let, for vhose coefficient functions are homogeneous polynomials of degree h. Denote [X 1, - ]h : H - Hh : Y --~ [X 1, Y ] and let Bh - Im ( [X 1, - ]h ). Annales de l’Institut Henri Poincaré -
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Choose for each h > 2
an
[R~
115
arbitrary supplementary space Gh for Bh in H~
(that is: Hh - Bh 0 Gh). Then there exists a germ of a C~ diffeomorphism such that ~,~X has an oo jet of the form
where gh E
h
Application (to
=
2, 3,
0)
-
....
be used later
G3 with Let X E 03
03C6:(Rn,
j1X(0)) (
=
on).
ax~ ~x,
a
O. If the z-axis 0.
012 0}2 R is
is
x
invariant under X, then there exists a diffeomorphism r~ : ([R3, 0) X’ has an ~ jet of the form such that
formally -
([R3, o)
=
,
Proof [Ta ]. Suppose, by induction, that for i E N, i X = X+
g2 +
...
write Rl Y E Hi with [Xi, Y]
We
can
>
1,
one can
write
with the gh~Gh and with ji-1Ri-1(0)=0. gi E Gi, jiRi(o) 0. Take an
+ gi-1 + R1-1 -1 - gi + b; with
=
bi. Taking §;= ~Y(., 2014 1 ) (the time -1 mapping of Y) one can check that + gi + Ri: (4)d*(X) == X1 + g2 + Since ji-1 ji-1 Id (0) we can apply the theorem of Borel (for diffeomorphisms) to obtain the desired diffeomorphism 03C6. =
...
=
Proof of the application.
For
forward calculation that for all p, q,
Vol. 3, n° 2-1986.
= ax ~ ax~ it follows from r
a
strai g ht-
16
P. BONCKAERT
We see that for each h > 2 the linear map natrix with respect to the basis
Hence Im [X 1, A basis for Ker
[X 1, - ]h
has
a
diagonal
[Xi, -~ Hh. [Xi, 2014 ] h is obviously Ker
=
So, applying the normal form theorem,
we
have for
4>*X
=
X’
an oo
jet
of the form
It suffices to prove that § can be chosen in such a way that it formally preserves the z-axis. Because then X’ also leaves the z-axis formally inva-
riant, and hence
cannot contain pure z-terms.
component the Y’ E Hi with
Take any [X 1, Y’== bi. If we separate the pure zi terms of o a the and components of Y’, we can write it in the form:
-
-
where Y E HI has
no
pure zi terms in
itsax~ and a components. We have ~y
.
d
as bi and [Xi, Y]don’t contain pure z1 terms in their 2014 component, ax necessarily A 0. So bi [Xl’ Y ]. As ~1 = ~ -1) (the time -1 mapping of Y) we obtain the result. 0 In chapter IV we will make extensively use of the following result. But
=
( 1. 22)
THEOREM
=
(Brouwer, Leray-Schauder-Tychonoff fixed point theoAnnales de l’Institut Henri Poincaré -
Analyse
non
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VECTOR FIELDS ON
rem) [Sm].
-
Let E be
a
locally
convex
[R~
117
topological
vector space, K a
nonvoid compact convex subset of E, f any continuous map of K into K. Then f admits at least one fixed point in K, that is : a point x E K with
/(.Y) =
~.
§ 2. One
The main result.
pose the
following question: suppose that a (generalized) formally invariant under a germ X E G3 (by formally we mean : up to the level of oo jets, or equivalently : up to the level of formal Taylor series); does there exist a Coo invariant one-dimensional manifold having oo contact with D ? The answer is yes, provided X is non-flat along D. This is one of the major ingredients of the following theorem. Let us emphasize that the results of this theorem are stated in terms can
direction D is
of germs in 0 E f~3. Let X E G3 and let D be a direction. If X is non-flat (2 .1 ) THEOREM. then there exists a Coo cone K of finite contact around D such that along D, one of the following situations occurs: -
I. all the orbits of X
hitting
K enter K and leave
K, except (of course)
{(0.0.0)~ II. D is
invariant under X ; there exists a direction D’ in K, D, which is invariant under X; if we arrange (by of X if necessary) that the orbit of X in D’ tends to 0
formally
contact with
having ~ changing the sign then either
A the only orbit of X in K tending to 0 is contained in D’ ; all the other orbits of X starting in K leave K ; B. all the orbits of X in K tend to 0 ; if we add 0 to such an orbit, we obtain a direction which has oo contact with D ; C. there exists a unique C° 2-dimensional subcone S of K such that
i ) S is invariant under X ; ii) all the orbits starting in S tend
to 0; if we add 0 to such obtain a direction which has oo contact with D; iii ) all the orbits of X starting in KBS leave K.
an
orbit,
we
Always assuming that along the invariant directions the orbits tend to 0 for t -~ oo, we find in cases II. A and II. C a unique model up to C° equivalence and in case II. B even up to C° conjugacy. The proof of this theorem will be given in chapters IV and V. (2.2)
PICTURES of the situations
Vol. 3. n° 2-1986.
occuring
in the theorem.
118
’. BONCKAERT
(2.3) re
REMARK. -
Perhaps sight.
in
some cases
the claimed results in
not evident at first
Take for
example
0
the linear vector field
1
(2.1)
0
+ -z2014 and X=x~x +y ~y time and have
) z-axis. All the orbits tend to 0 in negative ontact with the z-axis. Nevertheless one can find =
a
parabolic
of finite contact wo in this case) such that alle the orbits leave it, except of course the ne contained in the z-axis. This example indicates in which way the main leorem must be considered: once an orbit leaves the cone, we have no irther information about its future, which may be for example to tend ) zero or even to re-enter the cone. a cone
REMARK. - One can formulate and prove an analogon of theo(2 .1 ) in dimension 2, this time of course without situation II. C. On 1e other hand it is not yet clear to me how to generalize the theorem ) arbitrary dimensions; more specifically one could ask: does a formally mariant direction always hide a « real life » invariant direction? The mie question can be posed for diffeomorphisms.
(2.4)
m
Annales de l’Institut Henri Poincaré - Analyse
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VECTOR FIELDS ON
I~3
119
II. PREPARATORY CALCULATIONS CONCERNING BLOWING UP OF VECTOR FIELDS
Although the blowing up method for singularities of vector fields is known technique [Ta, Dul, Du2, D.R.R., Go ] we recall it here because it is of fundamental importance in the sequel. Especially for blowing up in a direction some specific calculations will be elaborated in full detail. Roughly spoken, blowing up a vector field is : write it down in spherical coordinates and divide the result as much as possible by a power of r (= Euclidean norm of x) such that one can take the limit for r - 0. a
§ NOTATIONS.
(1.1)
1. One We
-
spherical blowing
up.
denote, for m~N : 1m -+-1
So in fact with 03A6
we
introduced
Rm + 1 ) ( 0 }
on
a
sort of
spherical
coor-
dinates.
(1.3) PROPOSITION with X(0) 0. Then there is a Cp -1 field X =
on
Sm
If X
(or : ~~X = X). has flatness k, that is,
R such that in each q
x
C,(X(~)) =
e
S""
x
D =
0 and
~+iX(0) ~ 0,
then for the
following vector field -
we can on
S"’
take the limit tor
which
we
r
-
0 and
1 -
we
obtain
X is called the ( 1. 4) DEFINITION. X and up (once) diriding bij ~. -
Vol. 3, n° 2-1986.
a vector
held ot class
still denote X. vector field obtained
by blowing
120
?. BONCKAERT
X restricted to Sm x { 0} is a vector field tangent to Sm x {0}. The study of this vector field sometimes gives information on the asymptotic behavior of the orbits when they tend to 0 [Go ].
§ 2. One directional blowing
up.
In high dimensions the explicit calculations for spherical blowing up quickly become complicated. If we restrict our attention to one open halfsphere of S’" we use the better computable directional blowing-up. We can assume that the half-sphere has (0, 0, 0,1) as « north-pole ». x R by E Let us replace where E is an arbitrary normed vector...,
space. We do this because later on construction remains the same.
point
{0}
shall
use
this form and after all the
CONSTRUCTION.
(2.1) A
we
R.
x
(x, z)
~
of E will be denoted (x, z). With « the z-axis » we mean be the following map : T": E x R - E x ~: Let, for n E N, z). Similarly to proposition (1. 3) one proves
(2 .1.1 ) PROPOSITION. - Let X be a CP vector field with X(0) 0. Then there is a Cp-1 vector field X~1 on E such that in each q E E x ~ : if X(~’ 1 (q)) (or: X). Again jkX(0) 0 and we X 1 devide we denote the jk+1X(0) ~ 0, by resulting vector field =
=
X1
and
=
z
vectorfield
on
again we can
take the limit for
z
-
0 to obtain
a
still denoted XB
E
X1 is called the vectorfield obtained by blowing (2.1.2) DEFINITION. up X (once) in the z-direction and dividing by zk. Like in § 1, E x ~ 0 ~ is -
invariant under
(2.2)
RELATION
[R m+1 Let
so we can
consider the restriction
BETWEEN DIRECTIONAL AND SPHERICAL BLOWING UP IN THE
CASE.
denote the
insider the
ne
XB
«
upper
bijection
F:
hemisphere », that is : ST x R - Rm+ 1:i
immediately checks that ~ _ ~ 1
o
F.
Annales de l’Institut Henri Poincaré -
Analyse
non
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121
VECTOR FIELDS ON
Hence X
If
we
As
~Sm X ~
denote
and X1
F(xl ,
1
are
C~’
k is
(1+
...
r)
Z,
... ,
conjugated : FX =
an
... ,
xrn,
CALCULATIONS
(X 1,
... ,
Xm, Z) then,
the orbits
+ same
orientation.
CONCERNING ONE DIRECTIONAL BLOWING UP.
GENERAL FORMULAS.
(2 . 3 .1 )
=
X1.
everywhere positive function,
ofX~and F~X coincide. They also have the
(2.3)
r
=
write X (Xx, X~ ). It is an easy calculation to
If X is
-
a
vector field on E
x
[R
we
=
see
that
X if only if
/~
,
and thus rrom
now on we assume
(2 . 3 . 2) some
1HE
00 JET OF
X to be 1
X IN U
(CASE {Rm + 1).
(usual) multiindex abbreviations. m
For
then
we
denote
We write the oo
jet
ot X in U down as
if X has flatness k in 0. Vol. 3,~2-1986.
-
Let
us
indicate here
122
BONCKAERT
’or the
oo
jet of X1
Jsing (2 . 3 .1 ) i) a" -
and
in 0
we
write
calculating straightforward ,...,03B1m)
we
find :
for 1
m 1 _ x
i
for 1 i m 0 ii) a for iii) ~ 0 I n; i i ), ii), iii ) we may take the right hand side zero whenever it is undefined. =
=
(2. 3 . 3)
REMARK. - In
nd for each
a
with |03B1|=
in (iii) we see that for each 0. This expresses the fact that the
particular n :
c:
=
is invariant under X1or equivalently that f X 1 does not contain pure xa terms.
yperplane
§ 3. Successive directional blowing
the ~
z
=
0
component
ups.
~.1) Construction.
Let X1 be obtained by blowing up X in the z-direction and dividing by zk. Suppose that X 1 has again a singularity say in (a, 0) E E x { 0 } and that ~ 1 (a~ 0) = ~)_~ 0. Then we can blow up X 1 in (a, 0) in the z-direction in the following itural sense. Denote T : E x R - E x (x, z) - (x - a, z) ; since T*X is a singularity of flatness p in 0, we can consider the vector field T*X1 stained by blowing up T*X1 in the z-direction and dividing by zp.
i
DEFINITION. up X twice in the
(3.1.1) owing
T*X1 is called the vector field obtained by z-direction first in (0,0) then in (a, 0) and dividing
-
by z~ then by zP; we denote it X2. Of course X2 depends on the choice of the singularity of Xl. In this way, ~ can of course go on with the construction and define successively X" in some singularity. Obviously Xn depends on the I blowing up oice of the singularities in which we blow up. Let us formalize the idea ving a prescribed sequence of singularities in which must be blown up. ’st
_
Let X be a Coo vector field on E (3 .1. 2) DEFINITION. ;0) 0. A directed sequence of successive blowing ups of X is a such that triples (X~, (xn, 0), u{ X° X and i) (xo, 0) = (0, 0) ;
with
-
=
sequence
=
Annales de l’Institut Henri Poineare -
Analyse
non
linéaire
VECTOR FIELDS ON
0 _
ii )
N : Xn
n
has a singularity ot flatness kn m (xn, U) and
by blowing up X" in (xn, 0) and dividing by zkn.
is obtained
DEFINITION.
(3.1.3)
of X where dn, 0
A directed sequence of successive blowing ups N : (xn, 0) = (0,0) will for brevity be called a in 0 of X (in the z-direction). Such a sequence is
-
n
of blowing ups henceforth denoted by (Xn, 0, kn)o sequence
correspondence
n
N.
Further on we will show that there exists a 1-1 between directed sequences of successive blowing ups oo included). and N - 1 jets of directions (N
REMARK.
(3 .1. 4)
123
~3
-
=
(3.2) Calculations concerning
successive directional
blowing-ups.
We will give explicit formulas only for a sequence of blowing ups in 0. As we have already announced this is, at least for our purposes, no restriction. In this case we can define X~ in one step : LEMMA. 2014 Let Xn be obtained defined in (3.1).
(3.2.1) in 0 of X
by
a
sequence of
blowing
ups
as
== , ..,2014 ThenXnR. ~p~E
Xn where Xn has the property
=
X(03A8n(p)),
x
Proof
-
Straightforward.
D
(3.2.2) REMARK. - We see that properties of xn in a cylinder-shaped x [0, oo [ are transformed, by ~’n, to simular neighbourhood x [0, oo [. It is proporties of X in a cone of contact n -1 around { in this sense that we will obtain the results in the main theorem (1.2.1).
(3 . 2 . 3 )
FORMULAS. 2014 Just like in _
Further
on we
(3 . 2 . 4)
will need the
LEMMA.
/11
X
=
(Xx, Xz)
we
have
B
~~
following
Let X be
(2 . 3 .1 ) for
result.
Cx vector field on E x R. If X is non-flat x ~ is formally invariant, then there exists a Q E fvl and a C x function y: E with y(o, 0) ~ 0 such that + 1 the R-component of XQ + is of the form is the vector field z). obtained by blowing up X Q + 1 times in 0, in the z-direction without dividing by a power of z.)
along {OE}
x
[0,
-
oo
a
[ and if {
(XQ
Vol. 3, n° 2-1986.
.
124
P. BONCKAERT
Proof - Put X==(Xx, As, by our assumptions, the map z -~ z) has a nonzero jet, there exists a Q~N and C" maps Ao, Ai , ... , AQ : E ~ ~ with Ao(0) Al (0) == ... == A 1(0) - 0 and AQ(0) ~ 0 such that we can write =
Xz(x, z) Ao(x) + zA1(x) + The ~-component of XQ+ 1 is More explicitely : =
...
+
+
+
III. REDUCTION OF A SINGULARITY TO A SINGULARITY WITH NONZERO 1-JET BY SUCCESSIVE DIRECTIONAL BLOWING UPS
blowing up a singularity of flatness k one might hope that the atness of the resulting vector field in a singularity is not strictly bigger ian k. Up to one type of singularities, this will in fact be the case. If we consider a sequence of blowing ups (Xn, 0, we will see, Iso for that exceptional type of singularities, that after at most one step ve flatness kn becomes decreasing (perhaps constant). In fact, a vector eld which is the « blown up » of another one cannot be of that exceptional Ipe (mentioned above). Next we will prove that if the sequence kn does not decrease to zero, then ie vector field is flat along the z-axis. Let us start by showing that, up to a Coo change of coordinates, it is o restriction to assume that a directed sequence consists of blowing ups long the z-axis. When
§
1. Definitions and
properties
of directed sequences.
PROPOSITION. - Let N~N u { ~} and suppose that is a directed sequence of successive blowing ups of [hen there exists a C’ change of coordinates ~ : (p~m+ 1, 0) ich that in the new coordinates this sequence is ( ~,~(X)n, (0, 0),
0),
( 1.1 )
Proof
-
We define
inductively 1
on
as
some
1, 0) kn)o ~ ~ ~;.
transformations
follows.
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
!R 3
125
Put
Suppose
that ai,
’Ii
witn n + 1
defined tor 1
are
the singularity of ( ~n),~(X)n+ (~+i,0)e!R" (.Yn+i, 0) in the new coordinates induced by ~n’ Put
denote to
1
1
/
2014n~’ 1 -
/--
B
1
m.
We
corresponding ’B
1 ~mT 1 : (x, z) - (znx, z) denotes the « blowing up map », If qn : 1 N. n observe that ’Pn 0 Tn 03B1n o 03A8n, blown If Y E G~~ ~can be up n + 1 times in (0,0) and if ~n + 1 must be blown up in a singularity (an + 1, 0) then : =
.
~~
.
~
.
~ ~~
.
,
,
so must (the z component is not alterea by 1 n +1 nor by be blown up in (0, 0). we obtain that Applying the foregoing observation to 1 be is (which ( ~n + 1 )~c( 1 ) must blown up in (0, 0). For
(0)
’--
SO .
r or nmte
ØN -1
I
is tne aesirea
.
I
/~.....
i.,~
~).
by (1), consider the inverse limit of the ~", and theorem (I .1.12) of Borel together with the inverse function theorem [Di ] gives the desired ~. D For N
=
oo we
can,
For each directed sequence ( 1. 2) COROLLARY. (xn, 0), there exists a C~ change of coordinates (x, z) z’) and a direction D = ~ -1 (z-axis) such that if we blow up ~ * 1 X successively in 0 then points corresponding to (x, z) (0, 0) like in the proof of ( 1.1 ), are precisely (x’, z’) 0). Conversely, given a direction D = ~ -1 (z-axis), there exists a sequence ~ as above. Moreover, D is unique up to N -1-jet equivalence. In other words : there is a 1-1 correspondence between directed sequences and N -1 jets of directions. So we can speak of a sequence of blowing ups along a direction. -
==
=
==
Vol. 3. n° 2-1986.
126
.
BONCKAERT
If we blow up X successively along the z-axis it is of course possible hat after some finite time there is no singularity any more in (0, 0). Let us ~rmalize this as follows.
is called a maximal directed sequence 0, ( 1. 3) DEFINITION. >f blowing ups along the z-axis if either i) N oo ii) N EN -
=
_
Property. Such
a
sequence
always exists.
If (X",0,~)o~N is a maximal directed sequence PROPOSITION. vith N E N, then there exists a cone K of contact N - 2 around the z-axis ,uch that all orbits inside K enter K and leave K after a finite time.
( 1. 4)
-
Proof - As X~’~(O) 5~ 0, we can construct a cylinder shaped neighbourhood of 0 in [Rm x [0, oo [ of_the form C= {(~ R, : E [o, b [ } such that all orbits of XN - 1 inside C enter C and leave C after 1 finite time. Then
Let K be the germ in 0 of this set. D So from now on we will consider infinite sequences of blowing ups. Let us indicate what it means for X that a maximal directed sequence of blowing ups of it along the z-axis is infinite.
(1. 5)
PROPOSITION.
sequence of are
blowing equivalent :
Let ups of X -
along
be a maximal directed the z-axis. The following statements
i) N ii) the z-axis is formally invariant under X (see definition (1.1.20)). Proof i) ==> ii). Let us write down the oo jets as follows -
0 for all K > ko + 1 and i E Ve must prove that afo he formula in (II.2.3.2) ii) we find that for all K =
Annales de l’Institut Henri Poincaré -
>
ko
Analyse
+ 1 and non
linéaire
VECTOR FIELDS
i
1,....m}:
==
ON R3
and turther
aki,0
127
induction
by
on n we
get,
ato. using Suppose, by contradiction, that aKi,0 ~ 0. Then -k0-1 ~ 0 so k 1 + 1 K - ko - 1 (remember the choise of kin definition (II . 3 .1. 2)) 2 ; further by induction K - ko with always, as it should be, 0 kn _ 1 - n. kn _ 1 - n for some n. Certainly sooner or later 0 K - ko the
same
formula, for all n
=
...
=
-
...
-
m
X"(0) =
But then
ii)
~
i ).
ko diately ai,0 K >
Let
+ 1
~ 0, contradicting N
oo.
notations. We have ...,~1}. The formula (11.2.3.2) and f ~{ 1, ...,m} whence
the
us use
and 1, 0 for all n 0
=
the result.
=
§ 2. Reduction
to
same
a
singularity
with
nonzero
The main purpose of this section is to prove the
0, for all gives imme0, dn and =
1-jet.
following :
If is a maximal directed sequence (2 .1 ) THEOREM. of blowing ups of X E Gm+ 1 along the z-axis { 0 }m x [0, oo [ and if X is non-flat along the z-axis then there exists a N~N such that has flatness zero The proof of this theorem will be a consequence of the following propositions (2.2), (2.3), (2.4), (2.6), (2.7) and (2.8). D -
(2 . 2)
PROPOSITION.
-
Let X E Gm + 1 have flatness k. If X 1 has flatness n
As
0 ~x ~ k
=
1, 1
U we
i
~
only have
to look at the
a1 x 1
and the
1,
Out of (11.2.3.2) Üi) we obtain for each n k and a with = n -1: 0 with 1 _ n 1g = ci ~ ~ ~ ~ " ~ ~=c~ ~so~~ ~ =0, for each x with 0 ~ | 03B1| k. (2) From (I I . 2 . 3 . 2) i ) we get for each n with 1 n k + 1 and a with x = n and 1 xl : +
m.
=
But by 0 ~ x
(2) above those c’s ( _ k + 1 and 1
Vol. 3. n° 2-1986.
are zero.
xi.
Consequently
==
0 for each
oc
with
128
In the same way, using (11.2.3.2) ii), we obtain that all the coefficients in jk+ iX(0) vanish except possibly c~ ~1 with = k + 1. D
(2 . 3) PROPOSITION. has flatness
-
and if X~1 has flatness k, then X‘
If
k.
Proof Suppose that jk+ 1X2(4) that imply -
=
0. Then
proposition (2.2)
would
_
By our assumptions one of the ( = k + 1, must be different from zero. But this is impossible because of the remark (II. 2.3.3) (invariance of the z = 0 hyperplane). Q If XEGm+1 has flatness k and (2.4) PROPOSITION. then X 1 has flatness k + 1. -
Proof - We must show
and from jk+1X1(0)
=
0
we
that
~+2X~(0) 5~
0. From
proposition (2.2)
get
Because X has flatness k, there exists Using (II . 2 . 3 . 2) iii ) we observe that
an a
-k+2 - ~+2+~-~-1 _
# 0.
= k + 1 and
with |03B1| 1 .
D if (2 . 5) COROLLARY. - If sequence of blowing ups in 0 of X in the z-direction if X has flatness k then either.
is
(N
a
u{
directed oc }) and
i ) Xhas flatness smaller than or equal to k and for all n with 0 11 N -1I the flatness of Xn +is smaller than or equal to the flatness of Xn ; 1 ii) X has flatness k + 1 and for all n with 1 - n N - 1 the flatness of is smaller than or equal to the flatness of X~. Let X E Gm + 1. Suppose that there exists an (2 . 6) PROPOSITION. of blowing integer k > 1 and an infinite directed sequence (Xn, 0, -
X~ has flatness k and at each
ups in 0 of X in the z direction such that Vn
step
divide by zk. Denote i: R ~ Rm+1:z
we
-
(0,
...,
0, z).
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
is
II$3
129
Proof - Let us write again X for its formal part. Then jk- i(X)(0,..., 0, z) formally determined by
We
use
the notations
"
2014t~_2014
,
as
in
(II.2.3.2) and obtain 00
Hence the terms
n
playing
a
role in
o
i)(O)
are
1
we look at what these terms
give when blowmg up
We use qn:(Rm+1 ~ the formula in (II.3 .2) which becomes here m
For the
2014 component C,Y,
’
~
Vol. 3, n° 2-1986.
~--i
we
obtain
as In tne
proposition. and
30
a nd for
the - component we have
~ irst look at
(4). If this construction is possible for each
n E N then
Thus the second EE in (3) vanishes. For the same reasons as 0 for all N >- k + 1,0 z m, 0 - ~ tioned we must have Th is means 0 1 (X) f)(0) = 0.
c~
=
0
just men-
=
_
o
Let be non-flat along the z-axis. (2 . 7) PROPOSITION. Let xn be obtained from X by n successive blowing ups in 0 in the z direction. -
Then X~ is non-flat
along
the z-axis.
Let ~n : ~m + 1 ~ ~m + 1 ; (x 1 ~ ... , .xm ~ Z) -~ Proof z) the blowing up mapping leading to X~. Then for some analytic function Gn we have and X o ~I’n= ~~ X". Denote i : [0, oo[ ~- p~m + 1: z (0, ... , 0, z). Suppose that j~(Xn f)(0) 0. Then j~(Xn f)(0) 0 and hence j~(X f)(0) 0. But i = i. This would imply that j~(X i )(0) 0, contradicting 3ur assumptions. D -
...,
Je
o
-
o
0
=
=
0
0
=
o
=
Suppose that X E Gm+ 1 is non-flat along the (2 . 8) PROPOSITION. z-axis. Then it impossible that there exists an infinite directed sequence of blowing ups in 0 of X in the z-direction with Vn > 1: kn > 1. -
Proof - Suppose that there exists such a sequence. Then there exist integers k > 1 and N > 1 such that Vn > 0 : XN + n has flatness k. Hence proposition (2 . 6) ~)(0) = 0 contradicting proD position (2.7). This completes the proof of theorem (2 .1). o
IV. TREATMENT OF ALL THE FLATNESS ZERO CASES AND PROOF OF THE MAIN THEOREM, WITH EXCEPTION OF THE C° RESULTS
study germs in 0 of vectorfields which i ) leave the [R~ x { 0 } hyperplane invariant ii) leave the z-axis { formally invariant iii ) have a nonzero 1-jet. (Sometimes we will only consider the case m 2.)
We
=
Annales de l’Institut Henri Poincaré -
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131
VECTOR FIELDS
The
1-jet of such
a
vector field has a rA
matrix of the form
AI
m
with A E f~’~ " and where Cô has the same meaning as m (11.2. 3 . 2). Such vector fields appear as a result of the reduction by successive blowing up in part III. Even if the 1-jet is nonzero, we will sometimes blow up the vector field some more times in order to obtain situations like in the main theorem (1.2.1); also in some cases the proof of the existence of the Cx 1-dimensional invariant manifold is made easier by making extra blowing ups.
We must show that all possibilities for C6 and A (both not simultaneously a situation as in the main theorem (I .2.1). 0. This turns out to be fairly easy. We first distinguish in § 1 the case If C6 0, (§ 2) we restrict ourselves to the case m 2. The possibilities are :
zero) lead to =
=
i ) A is hyperbolic : see (2 .1) ii) A has eigenvalues ( « the rotation case » ) : see (2. 2) iii ) A has eigenvalues a, 0, a E tRB { 0 }: see (2 . 3) iv) both eigenvalues of A are zero : see (2 . 4). .
§
1. The
eigenvalues
in the z-direction is
nonzero.
satisfies : Suppose that ( 1.1 ) PROPOSITION. is formally invariant under X i ) the z-axis { 0 (see the notations of (11.2.3.2) or above). ii) -
Then
a) there exists a Coo germ h : ([0, oo [, 0) 0) whose graph (germ) I z E [0, oo [} is invariant under X and with j~h(0) 0 ; { (h(z), z) E Rm R b) supposing c10 0 change the sign of X if necessary there exists a cone K of finite contact around the z-direction x [0, oo [ such that the only orbit of X in K tending to 0 is contained in {(h(z), Z)E ~m ze [0, oo [} ; -~
=
all the other orbits
Proof ko ki1 =
a
starting in K leave K after a finite amount of time. If we use the formulas in (II.3.2.3) (with of course ... = kn-1 0) we find that for all n > 1 the 1-jet of X’~ has
-
-
=
matrix of the form
() l
Notice that 5. is
Vol. 3, n° 2-1986.
an
eigenvalue of A
if and
only
if
is
an
eigenvalue
132
P.
BONCKAERT
Hence for large n, X~ is a hyperbolic saddle for which we can apply the (un-) stable manifold theorem (see for example [H.P.S., Ke ]) to obtain the results. D
(1.2) REMARK. - The methods we develop further on for the delicate situations would also work here; the calculations would be much simpler here.
( 1. 3)
REMARK. - In
§ 2. From
(2.1)
The
( 1.1 )
eigenvalue
~3,
work in
now on we
THE HYPERBOLIC
we
may
replace
~
by
in the z-direction is
more even
any Banach space.
zero.
thus: m = 2.
CASE.
(2.1.1) PROPOSITION. Suppose X E G3 satisfies z-axis the is i) formally invariant under X -
3
8
iii )
a, b > 0
by
ii) X is non-flat
Then there exists
along the a cone
z-axis.
K of finite contact around the z-axis, a a COO germ h : ( [0, oo [, 0) -
C°2-dimensional subcone S of K and such
a) b) c)
that j~(0)
=
unique (fF~2, 0)
0 and :
S is invariant under X the graph of h, ~ (h(z), z)z E [0, oo [ }, is invariant under X and lies in S in K we have situation II. C of the main theorem (1.2.1).
8
Proof - By
lemma
(II.3.2.4)
we
may
assume
that the
- component
of X is of the form zQy(x, y, z) with y(0, 0, 0) # 0 and Q > 2. We may, and do, assume that y(0, 0, 0) > 0, because if not, replace X by - X which is of the same type. If we use the center manifold theory such as in [H.P.S., Ke] we obtain the existence of a unique center-unstable-manifold. Although it is general not necessarily CX, here we can use the uniqueness of the center-unstable manifold as well as the fact that X is non flat along the z-axis to obtain that the center-unstable manifold is in this case, as follows. For each In E N there exists a C7" center-unstable manifold Wm defined on some neighbourhood Um of 0 ; we can take care that U m +1 c Um for all By uniqueness of the center-unstable manifold we have Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
1
If
=
holds for fixed
we
take U 1 small
.
enough
then the
133
following
m.
Since ~(0,0,0)
>
0 and since a
>
0 there is a T > 0 such that
denotes the flow of X). So Ui1 n Wi is a Cm manifold. But as m is arbitrary, we obtain that U1 n Wi is C ~. The behavior of X restricted to this center-unstable manifold found in full detail in [D.R.R. ]. D
(4)x
can
be
The methods to obtain invariant manifolds, deve(2.1.2) REMARK. further on for the more delicate cases, could also be applied here. lopped -
(2 .1. 3) defined
PROPOSITION.
-
Let E be
Banach space, X
a
a
Coo vector field
with X(0) 0. Suppose that neighbourhood U of 0 E E the z-axis is invariant under i) X; formally is a hyperbolic contracii) E x { 0 } is invariant for X and D(X IE tion or expansion, that is : the spectrum of this linear operator is contained on a
=
x
in a subset of C of the form { z for some i~ e R, h > 0 ; iii ) X is non flat along the z-axis. iv ) X is bounded on U together with all its derivatives. Then there exists a cone K of finite contact around the z-axis and a C ~ germ h : ( [0, cxJ [, 0) - (E, 0) with a) the graph of h, ~ (h(z), z) z E [0, oo [ }, is invariant under X b) in K we have situation II. A or II. B of the main theorem (1.2.1).
Proof - Let us write X (Xx, XZ). By lemma (II . 3 . 2 . 4) we may assume =
’2014 ~- ~
,
0 and Q >_ 2. There exist A e Lc(E, E) and B: (E x R, 0) - (E, 0) such that
that
A,/~
XZ is of the form : ,
with 7(0, 0) #
and such that
(E
x
~, 0)
-
(Lc(E, E), 0) and Rx:
(up to changing the sign of X).
r Re z i ) the spectrum of A lies in { 0} ii ) B and Rx are Cx germs; especially B(O,O) 0 0) = 0. iii) =
is
This because the z-axis is formally invariant. We also may assume that R x flat along E x { 0 } because if not, blow up once.
Vol. 3, n° 2-1986.
x
134
P. BONCKAERT
The existence of the invariant graph follows from the center manifold Its smoothness follows from the fact that y(0, 0) # 0. If y(0, 0) > 0 then we have situation II. A and if y(0, 0) 0 then we have situation II. B; this can be proved by standard techniques like in [D.R.R. ] or like further on in the more delicate cases. D
theory.
(2.2)
THE
ROTATION CASE.
We study germs in 0 of vector fields on R3 which leave the z-axis R2 0 formally invariant and for which the 1-jet has eigenvalues here E Such a vector field is (/. fRB { 0 } ). completely non-hyperbolic, so the « classical » theorems about invariant manifolds as in [Ke, H.P.S. ] are not applicable. First of all, such a vector field has, up to a linear change of coordinates preserving x {0} and {0}2 x R, a 1-jet of the form -
In
[B.D.]] we obtained
the
(2. 2 .1) PROPOSITION. i ) the z-axis { 0}2 /
iii)
A is non nai
x
along
-
following
result :
Suppose X E G3 satisfies is
formally
invariant under X
x 1
me z-axis
then
a) there exists a Coo germ h : ( [0, oo [, 0) (~2, 0) whose graph (germ) { (h(z), z) e [0, oo [} is invariant under X and with jooh(O) 0 b) there exists a cone K of finite contact around { 0}2 x [0, oo [ such that in K we have situation II. A or II. B of the main theorem (1.2.1). -
=
(2.3)
ONLY
ONE NONZERO EIGENVALUE.
Here we will meet all situations II.A, II. B, II. C of the main theorem. In situation II. C we will have to construct invariant surfaces. Sometimes we could make use of the classical center manifold theory, but for other cases, where the requested invariant manifold is not a stable, center-stable, center, center-unstable or unstable manifold, we will have to apply our own methods. But if we set up the whole machinery anyway it can effortlessly be applied to the « classical » cases. The methods to obtain smooth invariant manifolds developped in this section, can also be used to obtain the results in (2.1) and (2.2). Annales de l’Institut Henri Poincaré -
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IF$3
135
Up to a linear change of coordinates preserving (~2 x ~ 0 ~ and { O}2 may assume that the 1-jet is of the form
x
R,
we
(2 . 3 .1)
PROPOSITION.
the z-axis is
i)
-
formally
Suppose X E G3 satisfies invariant under X
a
ii)
= ax
a ~ 0
X is non-flat
iii )
then there exists a germ h : ([0, oo [, 0)
along
the z-axis
K of finite contact around the z-axis and (~2, 0) with /~(0) 0 such that
cone -
a
Coo
=
a) the graph (germ) ~ (h(z), z) z E [0, oo [} is invariant under X b) in K we have situation II. A or II. B or II. C of the main theorem (1.2.1).
Proof. - We may assume that Because X is non-flat along the
z-axis,
and hence we assume that there exists of X is of the form
with
fl/(0, 0, 0) #
So X
can
in (I.1.21). apply lemma (II.3.2.4)
has the form
a
we can
Q e
as
N such that
the ~ component oz
O.
be written
as
follows:
where
0 i) fl~~~ 0) == ii) 0, o) ~ 0 i) S , is a vector field which is
oo
flat
along
the
z
=
0
plane
and with
c
zero
-,--component c~
(we absorbed
it in 7;
we
also assumed
at
least
one
blowing up) it;) Q > 2. We only pay attention Vol.
3,
n° 2-1986.
to the upper half space
[R2
x
[0,
oo
[. If we blow
136
P. BONCKAERT
up X n times, without dividing is nonzero, then we get
by
a
power of
z
of
course
since the
1-jet
;=)
0 plane. for some S~ which is oo flat along the z Because y(0, 0, 0) ~ 0 we can assume that X", denoted again X, has the following expression (new fl , y, ... ) provided that n is big enough : =
~-) /,(0,0,0)=0 it) /(0, 0,0) = ~ ~
0
P E { 1, ..., Q - 1}
~
is (X) flat along the z 0 plane and has zero ic) component. ~ V) y(0,0,0)=c~0. Furthermore, in order to make the calculations a bit simpler, we divide X ==
by
1 1 + -
The
a -
/
resulting
vector
-
field, again denoted X, is of the form
,
(new f; ... ) :
where f, y, S x satisfy the same properties (ff) to (v) as above. The foregoing manipulation is not essential. If we look back to the expression for X~ hereabove, we see that in case P Q - 1 we may assume that b. c 0 provided n is chosen big enough. So up to a change of the sign of X the four following situations can occur : =
I. II. III. IV. Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS
Here
we
already
theorem, II treat
that I will lead to situation II. B of the main situation II. C and IV to II. C. We We will refer to some lemmas which will be
announce
to situation II. A, III to
each case separately. after this proposition.
proved
In order to detect invariant graphs for X we consider the graph transformation defined by the time one mapping of X (precise definitions will follow). Roughly spoken this is: take a graph of a map (x, y) h(z) and transform it by the time one mapping of X. First we modify X, without modifying its germ in (0,0,0), with a Coo =
« bump » function r : [0, on [1, oo [ as follows. For all 8
>
0
we
oo
[
-
[0,1 ], where i(u)
=
1
on
-0, -
-
and z(u)
=
0
define / 7X
Then XG has the same germ in 0 as X can be defined on a neighbourhood of the form (x, [0,00[, with B(0~)== { (x, y) E }. Moreover the z)| z ~ 8 } is invariant under - X~. We can modify X to XG in such a way that the (new) f, y satisfy satisfy inequalities like ~
-
r,__
-.
~B
~
~
-
A
neighbourhood B(o, ,u) x [o, oo [, where al, a2, bl, b2 can be independent of E (for this reason we don’t attach an index E to f or y).
on some
chosen Let
us
show that the
«
fiatness condition
»
of the
is independent of E, that is : for all r, S E N there exist b such that for all E > 0 and for all (x, y, z) E B(o, ~) x 1
Let in tact r,
Vol. 3. n° 2-1986.
,.
i
i v
W e have :
, ,
E
term 03C4(z ) >
S x (x, y, z)
0 and
[o, 3] :
>
0
138
P. BONCKAERT
There exists a ð > 0 such that for all p E ~ 0, there exists a constant Li,p > 0 such that
...,
,
- ~ --
z
The time
1-£)= mapping
~ ~:
one
z) =
(1
0
+ s ~ and all i E { 0, B(o, /1) x [o, ~ ] :
...,
s}
,
Furthermore for all j e fil there exists I T~(M)) ~ K~ . Hence for all z 6:
and for all
r
on
so
the
of xE
can
+
y,
a
K~
>
0 such that for all
[0, oo [:
Me
problem is trivial there. be written down
z))y, z
+
y,
as :
z))
+
y,
z)
where
i)
is oo-flat
along [R2 x { 0 }
and the flatness condition is
indepen-
dent of 8
has zero z-component (we absorbed its z-component in g) ; ii ) iii ) there exist constants b2 and a neighbourhood
independent
of ~, such that
on
V:
We write Fe We will show in lemma (2.3.5) that there (Fx,e, Fy,e, > 0 and 03B4 E ]0, 5i[such that if h : [0,5]] - [R2 satisfies h(0) 0 and for all ze [0~]: [ h’(z) ( [ 1 then Z : [0, 5]] z z) is a diffeomorphism onto (at least) [0, ðl]. Let z(Z) denote the inverse. We define, for 0 8 61, the following =
exist 03B411
=
function spaces :
[0, ð] -+ [R2
F?,. = ~ h
is Cm,
h( [~~ ~ ])
c
B(0,
1, h ~ - o
and for all
[0,5]]
-
[R2 is Cm, h(0)
I Define H
=
as
H(Z) =
=
0} [s,5]: ~(z)==0} i E ~ 0, m~~. ze
[a,3]: h(z)
=
0 and for all z E
[
for all
z(Z)),
...,
z(Z))) if Ze[0, ð1]]
Annales de l’Institut Henri Poincaré -
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ana
=
U lI L E
L~$
139
is a wen defined
1E
map
iium
into
tsE-
is small and Fg is continuous for the C"" topology. From lemma (2.3.6) we will obtain that for all m > 1 there exists an c Fm. E > 0 such that for the Let Fm be the closure of F7 in topology. Each Dth for h E Fm and Lipschitz function with a Lipschitz constant independent of h ; this is still true for h E Fm. The space Fm is hence compact for the topology, as a consequence a theorem of Ascoli-Arzela [Di ]. of F~ is also convex. =~ As F is continuous on F~ for the topology, we still have
provided 8
Fr. Hence it follows from the Leray-Schauder-Tychonov fixed point theorem (1.1.22) that F has at least one fixed point in F~. We hence obtain for each m an Em > 0 and a C"" map hm: [0, 6 ] - [R2 with jmhm(0) 0 whose graph is invariant under =
z) (0, 0, 0) hm(z) 0 on [~m, 03B4 ] and because lim in z the estimates we is on obtain ceo outside 0. uniformly (see g) that hm In lemma (2 . 3 . 7) we will prove that if h : [0, 5 ] is C1 on [0, 5] B(0, p) and COO on ]0, 5]] and if the graph of h is invariant under then h is ceo 0. on [0,5]and jh(O) Hence hm will be So we can consider for example F~and hi. As a matter of fact we want hl to be invariant under XE1. To see this, observe that for all t E [0, 1 ] : ~xEl((0, £5), t) lies on the z-axis denotes the flow of Xe1) since ~1 5i; for all t’ > 0 we write t’ = n + t where t E [0,1 ] Because
=
=
-
=
and n
we
have I
~~~
C~B
t
tB
/
t
///1
C*B
B B
B
nence me
grapn 01 ni is invariant under 03C6x~1(.,J tor an t’ > u. in Finally lemma (2 . 3 . 8) we show that in some neighbourhood of(0,0,0) all orbits tend to (0,0,0) and that those outside [R2 x { 0 } (in the blown up situation) are graphs of maps h satisfying the hypothesis of lemma
(2.3.7). Hence
we are
in situation II. B of the main theorem
Here the graph transformation can be defined mapping of X is again of the form
where
is oo tlat
Vol. 3. n° 2-1986.
along
the
z
=
U
plane
as
(1.2.1).
follows. The time
and has a zero
one
z-component.
140
P. BONCKAERT
and
This time there exist constants (0, 0, 0) such that on V we have ~-
~
Here
we I" -
-
_
0 is
8 >
_
-
_
(Fx, Fy, sufficiently
.,
_
~m
~
"
,
n
--~-
small and if h E
~
then the map
(at least) [0, e]. To
~
~
.
~
prove
D20141/7/ B B .
this, remark first that
/
estimate the first order derivatives of of h E F~. Second observe that for small
we can
independent
Denote
~.~
=
is a diffeomorphism onto for small z:
because
..
V of
take the function spaces
Denote F
If
../~
neighbourhood
a
the inverse of Z ; put H
rtB
g and
h
by constants
s:
I1 h where
H(Z) = (Fx(h(z), z), Fy(h(z), z)). Again : i ) r : Fm B~ is well defined if 8 is small and r is continuous for the C"’ topology c ii ) for all m > 1 there exists an 8 > 0 such that proof see lemma (2. 3. 9) hereafter iii ) r has a fixed point in Fm (the closure of Fm for the topology). z
=
-
_
In order to prove that hQ is Cx we will show in lemma_(2.3.10) that there exist > 0 such that is a sequence in B(0, p) x [0, s] ~Z, with lim and for i ~ 1 (0, 0, 0) (xi-1, Yi- 1, y,, z,) y,, z; ) then this sequence must lie on the graph on hQ. This implies that for all m > Q there exists an ~’m > 0 with ~’m ~ min { 8m, 8Q } such that h,~ and hQ coincide on [0, s~ ]. As a consequence of the movement of F in the z-direction (see the estimates on g above) we see that hQ is C7" 0. Hence hQ is Cy and jx h(0) on [0, GQ]] and that 0. Because of the unicity of hQ. in the sense of lemma (2.3.10), we see as follows that the graph of hQ is also invariant under X. Let q ~ N/{0}. Then, by similar arguments as above, we find an E E ]0, 8Q [ and a C x map h : =
==
=
=
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS
[0, I ]
---
fR2 with jx h(o)
(the time 1 q must.
by
mapping
lemma
of
(? . 3 . 10),
=
0 whose
X). lie
Take
on
the
ON R3
141
graph is invariant under 03C6x(1 q,.) z E
[0, ~].
Then the sequence
graph of ,
(l )
So h hQ z ] . Hence the graph of hQ is invariant under ] q / for each q. Thus under X.. lemma it will II. A from follow that we in situation are (2 . 3 .11) Finally of the main theorem. ==
Case
I I I : P Q - 1, ~ 0, b > 0,
,
c > 0.
Let us first search for an invariant surface tangent to the yz-plane. This will be a so-called center manifold. If we would apply the « classical » center manifold theorem, we would obtain for each r the existence of a C7 center manifold : see for example [Car, Gu, H.P.S., Ke, M.M. ]. But we will show that in this particular case the center manifold is Coo, at least in a blown up situation. Further we will show that the orbits starting in some neighbourhood of 0, but starting outside the invariant surface, will leave the neighbourhood. The behavior of X restricted to this invariant 2-dimensional manifold tangent to the yz-plane is well known and is as follows: there exists an invariant direction D’ having oo contact with the z-axis ; all the orbits starting in the invariant surface tend to 0 in negative time and have oo contact with D’. So we will be in situation II. C of the main theorem. But let us now come to the proof of all this. First we prove the existence of the invariant ex surface. For reasons which will become clear later in the lemmas we first perform a rescaling of the z-axis as follows. We want in fact that P > 2. Let R : [R2 x ]0, oo [ - [R2 x ]0, oc [: (x, y, u) - (x, y. u2). We have that R*X’ == X if and only if for
Vol. 3. n° 2-1986.
142
Here in
P. BONCKAERT
particular :
instead 01 M, A instead 01 X .
Let us write again
z
We obtain
vector
again
a
field of the form
(new P, Q,f, ...). ~
~
~
with this time P > 2 and still P Q - 1. Let us already remark here that such a rescaling will have no influence on our result, since it is our aim to construct an invariant graph of a Coo function z) with for all y: jxh(y, 0)=0: it is easy to show that then also a C x function of (x, u) oo-flat along the y-axis. Also all the other results we shall obtain are preserved by this rescaling. The time one mapping of X can be written as
~==/!(~~/M) is
F(x, y, z) = (eax, (1 + where a(0, 0, 0) > 0, g(0, 0, 0)
z))y, z + zQg(x, y, z)) + 0, Roo is oo flat along the z
>
=
has zero z-component. There exist constants al, a2, such that on V:
bi, b2 and
n / n
We look for
a
Ceo invariant ,.
~..
,
,
*~
>
graph ...
I
,
,
’7
a *~
z) 0 plane and R
neighbourhood
V of
~
(0, 0, 0)
n-
of the form -
,
,,
> 0 will be rechosen some times for our purposes. If , 3 > 0 c V. small then VÓ,Jl:= [- /1, p] x [-~/~]] x [0,5] ci V and Let us define some function spaces for ~ ~ 3 :
The
are
and FI = Fx, F2 Let us write F (Fx, Fy, (Fy, F,). We would like that the surface determined by ( y, z) - F(h( y, z), y, z) is again the graph of some function. For that purpose we shall prove in lemma (2.3.12) =
=
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143
[R~
sjJ small /1, 8 > 0 and h~F10,~ the map (Y, Z) : [- , ] [0, onto a is ~c ] x [0,8]. /1, (at least) [ diffeomorphism F2(h( y, z), y, z) ( y, z) In that case we can take the inverse of (Y, Z), and denote it ( y, z). So ( y, z) is defined on (at least) [ - ~c, ~c] x We put H rh where -
that for
-
=
H is defined on (at least) [ - /1, /1]] x [0, e]. So, for 8 small, r is a map defined on and takes values in B~. r is Moreover continuous for the em topology. In lemma As Fm c for small 8, r is defined and continuous on we that there a shall exists > that 0 such for each (2. 3 .13) prove (fixed) ~ > 1, there exists an 8 > 0 such that c Fm. From this fact, from the theorem of Ascoli-Arzela and from the Leray-
Schauder-Tychonoff fixed point theorem (I.1.22) we derive, in an analoguous way as in case I, that for all m E N there exists an 8m > 0 and a C"" map R with jmhm(y, 0) = o for all YE [ - , ] whose graph hm : [- , ] [0,~m ] is invariant under F. In lemma (2. 3 .14) we shall show that there exists 0 such that if (xi, yi, is a sequence in [ -,u, ,u] x [ /1, /1] x [0, 8] with and F(x;, yi, 1 ) then this sequence must lie on the graph of hQ. Again this uniqueness of the invariant graph and the movement in the z-direction (see the estimates on g) imply that hQ is in fact ceo and that its graph is invariant under the vector field X. Also j~hQ(y, 0) 0 for all y E [ - ,u, ]. -
-
=
Next we consider the restriction of X to this invariant C°° 2-dimensional manifold obtained above (always restricted to the upper half space). This 2-dimensional situation is treated in full detail in [D.R.R. ], and is hence omitted. Finally, from the expression of X we can now easily show that we are in situation II. C of the main theorem (1.2.1).
Here
we
also look first tor
an
invariant
surface, this time tangent
to
the xz-plane and passing through the x-axis. In contrast with case III this surface is not a center manifold. But nevertheless we will prove that we are in situation II. C of the main theorem, that is: the orbits starting outside the invariant surface and starting in some small neighbourhood of 0 leave this neighbourhood, on the other hand the orbits in the invariant surface just mentioned tend to 0 in negative time and have Jo contact with the z-axis. One can observe in the sequel that in this case it is crucial that the xz-plane is formally invariant, thanks to the normal form. If this were not the case, our method wouldn’t work. Vol. 3, n° 2-1986.
144
P. BONCKAERT
The time one mapping F of X has the same form as in case III with this time (x(0, 0, 0) 0 and g(0, 0, 0) > 0. Hence there exists constants al, a2, bl, b2 and a neighbourhood V such that on V : 17
If
0 are small then F(V,,,) V. We define for E 3 :
/1, ð
>
V J-l,ð
~==
[- /1, /1]]
x
[-
]
x
[0, (5]c
V and
c
Again: see i ) for small 8 we can define the graph transformation r on lemma (2. 3.15) here after. B~ is continuous for the Cm topology; ii ) there a (fixed) ,u > 0 such that for each m E exists >- 1, there iii ) exists an 8 > 0 such that Fm : see lemma (2 . 3 .16) ; iv) applying the theorems of Ascoli-Arzela and of Leray-Schauderthe existence of an 8m and a C"’ map hm : Tychonoff yields for all m 0 for all x E [ - ~c, ,u] whose [2014~,~]] x [o, Em ] ~ R with graph is invariant under F; v) the uniqueness in the sense of lemma (2.3.17), and the movement of F in the z-direction (see the estimates on g) imply that hQ is in fact Cx and invariant under the vector field X. 0) 0, for all x E [ 2014 ~ ,u] ; vi ) the results of [D.R.R.]] concerning the behavior of X restricted to the invariant surface together with lemma (2.3.18) hereafter imply that we are in situation II. C of the main theorem. D =
=
Lernmas used in the
proof of proposition (2 . 3 ..1 ),
(? . 3 . 5) LEMMA. 2014 There exist 6 > B(0, p) satisfies h(0) 0 and !! [0, 6] -
0 and
case
I.
a 1 E ]0,5[[ such that if h : on [0, 5]then for all 8 > 0
1
=
the map v.
is a
diffeomorphism
1
_
onto at least
FTD . -,
. 17
/ L/ ~B-B
[0, 03B41J
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS
Further Z’(z) > ð11 = 5 + D (independent of h). Put
LEMMA.
(2 . 3 . 6) such that
Proof we
+
0(zQ)
>_ 21 for small ð
1, there exists
an 8 E
]0,~i[[
Fm.
c
-
QzQ-1b1
1 +
For each
-
145
Let h E
Fm and put
H
=
reh.
For
z
=
z(Z) and (x, y)
=
h(z)
have
0 and P Q - 1 and a 0. induction for all on i, . that Suppose, by II and Let us abbreviate ... , F1 h(z) := (h(z),z ) :=(Fx~E, j E ~0, 1, 2014 1 }. and F2 :== times and obtain for We differentiate the equality the left hand side, using the higher order chain rule [A.R., Ya ] :
since a2
_
_
where
is some « universal summation over
C10)
properties jl
+
+
...
jk
isolate the term with k :>
(F2 h))(z) :>
=
=
i and
jp
>
1 for all
WIth tne
Let
us
i in the summation : :>
=
...,7~
/?e { 1, ...,~c}.
h)~Z)~
...,
i- 1
There exist C x functions
D’(H (F2 ~))(’) ç
0
=
Ak, k E ~ 0,1, ...,i DIH(FZ(h(Z))) - (D(F2
ç
;-1
Vol. 3. n° 2-1986.
- 1
},
such that ...,
we can
h)(z))
write
146
P. BONCKAERT
For the
right hand side we get
Intermezzo:
If E is a normed space, if L : E ~ E is an invertible SUBLEMMA. E is a continuous i-linear map then continuous linear map and if B : -
I
where B
(L,
...,
L)
I -r.. "
-
I -
T ’I. I 1
/T
is tne i-linear
map
1
1 T - ~1 II 7
aennea
oy
~’roof : 2014 Ubvious./. If
Let
we
us
apply this
make
an
in the
equality
estimate tor each term or tactor
separately
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
As
we
already
saw
in lemma
(2 . 3 . 5) :
symbol, as well as
Remark that this 0
R3
147
,
the
following,
ones
are
independent
1») because of the induction hypothesis ; this 0-symbol is Dih(z), by a independent of h since we can bound the II h E of constant independent F;’. =
...,
2. + 1k = In each term ot this summation we have j1 + So there must be 1, 2, ... , k ~ with jp f - 1. Hence, except for the case that ji1 jk = 1 and k = i, each term contains a j2 1 ~)_ factor Dkh(z) with k E {0,1, ...,f - 1 }; such a term is Suppose that ji = j2 = ... = jk 1 and k i. We can write ...
=
=
=
...
=
as a sum
of terms
containing Dh(z) as a factor, except the terms ot the torm
if
which is
=
we
replace (x, y) by h(z)
w e nave :
So if
write h =
(h 1, h2) ~ ( DF’Oh(z)) - h~i~(z) ~ ~_ ~ ~ we
Vol. 3, n° 2-1986.
we
get, since (x, y)
(1
+
~~x~ Y~
must be
replaced by h(z) :
148
P. BONCKAERT
e) Finally We end at : il1 i
because P
I
Q -
(2 . 3 . 7) LEMMA h(0) 0, if
1
0.
and a2
. - If h :
[0,5]] +
=
[0,5] Proof
on
D
00
-
B(o, ,u) is invariant under on ]0, ð] ] then h is
and if h is
if C x
0.
=
For all Z E ]0, ð]] we consider the sequence The invariance of the graph of h implies that we can write -
z. for alli E N, with in particular zo(z) z E ]0, zb] is of the form zi(z) then If Za E ]o, ~] and zb each zl(za) for some z E In order to prove the lemma it hence suffices to show the following: > 0 such that for all zo E [Zb, za ] for all j, s there exist za E ]0, 5 ] and and all i =
=
II
we can abbreviate this
shortly
zi
B B II ----
by writing
Let j,
=
From the
~~/1)/-/-
se
B1
>
=
Let us write
formula
recurrence
on
F~m we derive the existence of za
>
U
then for alli >_ 1 :
0 such that if
lor small za since a
/-/-
~J.
tor alli > 1 and from the estimates
and
v
u.
Applying this successively
we
get
~-11
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
[R~
149
On the other hand
II
," I
If
then for
Q - 1 ).
P Thus in that
(remember :
case II
1 /
B ) )
Choosing
we
have for
can find for all f =
We
constant
a
Ls such that for all
ze
[zb, za]:
~n(z)~ ~ Ls. ~~
t-1
From
+ t-i1
Vol. 3. n° 2-1986.
~2~ ~)
we
derive 2014~~~~ ~ "
so
So
150
P. BONCKAERT
We obtain :
for all i E Thus the case j 0 is done. Now by induction on j we prove that differentiate the equality =
for all
=
We
s e
j times (where h(z) _ (h(z), z)). Proceeding for that like in the proof of lemma (2 . 3 6) -
~
!!’B
. ~
_.
_._
we
find for the left hand side :
_..
__
where Ak is some ceo function; it is important to observe that Ak is linear with respect to the variable For the right hand side we get :
where the terms in the last summation +
oo we see I ir.
never
contain
Since
that
i_m _
’B I -
i
~
1I
AI - 0 - 1B
exactly like in the proof of lemma (2. 3. 6) which a priori might be unbounded ; which is, by the induction hypothesis, an
For the other terms we proceed but collect the terms containing
those terms contain a factor for any N E fil. Also -
So
we
-
_
n
v
.
T’Bo
v
_i v
v
find that tor all
Choose N
=
Q
+ s ; then we can find constants al
0 and B
j
>
0 such that
Annales de l’Institut Henri Poincaré - Analyse
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VECTOR FIELDS ON
This estimate is of identical type the case j = 0. Now we may go obtain the desired result. D
as
the
on
151
started from when solving in the same way in order to
one we
exactly
(2 . 3 . 8). LEMMA. - There exists a neighbourhood V of (0,0,0) such that all orbits in V tend to (0,0,0) and those outside 1R2 x { 0 } are graphs of maps h satisfying the hypothesis of lemma (2. 3 . 7).
Proof - By means of the obtained Coo invariant graph f (h(z), z)z E [0, ð] ] ~ define the Ceo coordinate change
we
....
Put X G*X and write X riant we observe that =
From the
-
Since
Xy,
=
_
X
leaves the
inva-
expression of X we calculate that, for the standard inner product
[R2:
on
-
Because G does not alter -~
~ "-
......
oo
-
-
jets in (0, 0, 0) we can write for X : -"’"
_n
~.-
-.
-.
~
.
~
.20141
b 0 Let us write again (x, y, z) instead of (x, y, z). Since /(0, 0, 0) a and constant a we V of and since a can find 0, (0, 0, 0) neighbourhood c > 0 such that on V : =
~
^-
From this the result is Lemmas used in the
(2. 3 . 9) that
LEMMA. c
~......-
...... _
easily derived.
D
proof of proposition (2 . 3 ..1 ),
-
For each
>
case
II.
1, there exists
an 8 >
F§l’.
Proof The differences with the proof of lemma (2 . 3 . 6)
for
b) :
Vol. 3. n° 2-1986.
are :
0 such
152
P. BONCKAERT
for
e) :
LEMMA. There exist ~,~ > 0 such that if x in (0, 0,0) and j~, zi) B(0,~) [0, £ ] with lim sequence
(2.3.10)
is
-
a
=
.
graph of CQ coordinate change
then this sequence must lie
Proof -
we
With the
obtain for F
=
on
(Fx, Fy, FZ)
the
:=
G*F
that
0 on some neighbourhood of(0,0,0). for some a 2 Denote the transformed sequence (Xi, yi, Zi)iEN. We haveI ( xi + 1, Yi + ~ ) II if 8 is small. So then for all i : 1 1 1 .. ~ _ IIl 1 ~ 0 form -~ oo. Thus Xi Yf= 0, that is: the sequence lies on the z-axis, which is the transformed of the graph of hQ by G. D
( x_i,
=
(2 . 3 .11 ) LEMMA. - There exists a neighbourhood V of (0,0,0) on which all orbits of X starting outside the invariant graph leave V for t
-
-
Proof and
on
oo.
Precisely like in lemma (2.3.8) some neighbourhood V: -
~ rom this s a
we
obtain for
some
B
that the function V : ~3 -+ [R: (x, y, z) -~ ~(x, > 0, whence the result.
we see
Lyapunov function for X in the region z
emma used in the
proof of proposition (2 . 3 .1 ),
case
>
0
y) ~ ~ 2 D
III.
the map (Y, Z) : (2 . 3 .12) LEMMA. - For small /1,8> 0 and h E R2 : (y, z) F2(h( y, z), y, z) is a diffeomorphism onto [0,8]] - , ] at least) [2014~,~]] x [0, 8]. -
-
Annales de l’Institut Henri Poincaré -
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F’2~~ Y~ Z)~ Y~ z) _ « 1 +
z)~ Y~
Y~
where Roo is 00 flat along the So, in a short notation : ~
-
We have:
153
+
Y~
and has
z-plane
T.~
..,.
)R~
~ _...
Z)~ Y~ z)) +
zero
~--
Y~
z)~ Y~ Z)
z-component.
D1
D"",
D ~ ........
W
D(Y,Z)(0,0) Identity. Because of the inverse function theoand because for all h E > 0 such 1 there exist [[D h(y, that for all h E the map (Y, Z) is a diffeomorphism on [2014~,~] x [0, a]. Put Roc 0). If 8 is small enough we have for all y E [- ~c, /1] : =
rem
=
C
and tor all ze ~it
As
there exists
Proof -
..
-- B
~B
~~ P
.. n
x
is
simply connected,
LEMMA. - There exists a J1 an 8 >
We
0 such that
=
rh. For i )
ims
t
impnes
1
> U such that tor all
c
(2. 3 .12) holds. and ( y, z) ( y, z)(Y, Z) we
such that lemma
choose ~
Let h E F~, H
T
Suppose, by induction
lUl
m
~U, 8j:
...Pwi~ I ~ .
(Y,z~[2014~~jJ
(2 . 3 .13)
~~
17
=
0
r’""7"B. II
on
1, ... ~2014 1- J
=
II
B
n7 i
)).
have:
AI nDB
i, that
1
tlllU
=2014
1.
A ’ LJ1
brevity put
and put also Vol. 3, n° 2-1986.
Fi1
=
Fx, f’2 =
It
we
differentiate the
equality
154
P. BONCKAERT
H
i times then
F2 0 h (2.3.6) that
=
We estimate
as
we
obtain in the
same
way
in lemma
as
follows :
h)( y, z) expression for a) From we see that we can write (2.3.12) the
=
D(Y, Z)( y, z) obtained
lemma
where Mh( y, z) is some matrix with M~(0,0) 0 ; this because Q see the rescaling construction. Moreover we can choose > 0 such that for all m E and all ( y, z) E [ - ,u, ,u ] x [0, e]: =
-1 > P > 2 : m
>
1, all
Then
because of the induction on the llAk(h( y> Z» > y> ...,
hypothesis z) II.
and because of the uniform bounds
1
because here the same here in particular
reasoning
as
in lemma
(2. 3. 6) applies and because
for
each q E ~0, ...,~ } d ) From tnnales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS ON
we
1R3
155
obtain
e) r many :
ii s is small we nave ior an t
E l u, ... , m j :
~~a .I
I
n/-11; -
Hence
(2 . 3 .14) LEMMA. - If E is small and if (xt, y~, [2014~,/~]] x [- /1, /1]] x [0,8]] with lim (~,Zf) (0, 0) =
v
T2014’/
is and
a
sequence in
B/’~~B v
/
Proof. Similar to the proof of lemma (2.3.10) with this time the coorchange -
dinate
r
Lemmas used in the
(2 . 3 .15)
is
a
~- 20142014 TT 20142014 ~~Yl7 7~
proof of proposition (2.3.1),
LEMMA . - For
sufficiently small 8
diffeomorphism onto at least [2014 ~ /1]] We have (X, short notation :
Proof so, in
a
1 for
As [[
independent
of h.
Vol. 3, n° 2-1986.
z)(x,
z +
x
>
case
IV.
0 and for h
E
the map
[0, 8].
h(x, z), z)) + R ~(x,
remark that the last matrix is
z), z)
an
156
P. BONCKAERT
ea
D(X, Z)(0,0) =
We have:
0
.
Because of the
remark
foregoing
> 0 independent and the inverse function theorem there exist small is a diffeomorphism. such that (X, Z)[ on For all x E [- /1, /1]: 8 + EQg(x, h(x, 8),8) > 8 if ~ is small because of the estimates on g on V. + Also for all ze [0, s] and 8 small h(,u, z), z) > /1 and X
~(- /1)
h( - ~c, z), z)
+
-
,u.
Hence, since (X, Z)( [ - ,u, ~c ] x [0,6]) is simply connected,
LEMMA.
(2.3.16) m >
1, there exists
Proof
a)
We
an
Copying
-
We put F
-
=
can
There exists a J1 8 > 0 such that
0 such that for each
>
c
the scheme of lemma
(Fx, Fy, F )~ Fi = Fy~ FZ
=
(2. 3. 6) this
time
one
has :
F~).
write :
hence
b) and c) : d ) here :
the
same
so
for small
z
since
0 and
bi
>
0.
D
Annales de l’Institut Henri Poincaré -
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157
VECTOR FIELDS
~~ . ~ .1 i ~J
LEMMA.
-
11 8 is Small ana 11
lim
yi,
IS a
sequence
ill
and
(Y~~ z,) (0, 0) F(x;, y~~ z,)=(~-i, y~~ Z-~) [0, s] with [ - p, /1]] > the then must lie on this (i 1) sequence graph of hQ. the to of lemma 0 proof Proof - Analogous (2.3.10). x
(2. 3 .18)
LEMMA.
-
which all orbits of X t
-
=
There exists a neighbourhood V of (0,0,0) on outside the invariant surface leave V for
starting
DC.
-
Proof
- After the coordinate
change
r -
-
obtain a vector field X GX whose y and form (write again x, y, z instead of x, y, z) : we
:=
,
D
.
anu
wnerc j
Take a bounded such that on V :
are
of the
~.. ,
neighbourhood ~l
V
of(0,0,0) and constants a2, b 1, b2 > 0 -
Now the lemma easily follows. D This completes the proof of proposition ALL
components
u.
v
(2.4)
z
(2.3. 1).
THE EIGENVALUES ARE ZERO.
Here we will meet all situations II. A, II. B, II. C of the main theorem. Since we work with germs X of flatness zero, that is: 0, and since the z-axis must be formally invariant under X, we can assume, up to a linear change of coordinates preserving [R2 x {0} and {0}2 x [R, that the
1-jet
lines of this
is y~ ;equivalently: the matrix A introduced in the first c.~
chapter
IV is:
fo ana
Co
=
u.
Vol. 3. n° 2-1986.
it
158
P. BONCKAERT
of the previous and (2 . 3 .1 ). With (2 .1.1 ), (2 .1. 3), (2 . 2 .1 ) « reducing » we mean : to deform the vector field by means of (sometimes degenerate) coordinate changes which do not alter the nature of results such as for example « having oo contact », « being a cone of finite contact », etc. In fact all the cited cases will occur. We will try to reduce, in in propositions ( 1.1 ),
sense, this
some
case
to one
cases
(2 . 4 .1 ) PROPOSITION. Suppose X E G3 satisfies i ) the z-axis { 0 } 2 x [R is formally invariant under X -
ii) X is non-flat
Üi)
along
the z-axis
then
[R2 whose graph (germ) a) there exists a Ceo germ h:([0,oo[,0) z oc E is invariant under X and with j~(0) 0 { (h(z), z) [0, [} a exists cone K of finite contact there b) around { 0}2 x [0, oo [ such that -
=
in K
we
have situation II. A, II. B
or
II. C of the main theorem.
lemma (11.3.2.4) of X is of the form
Proof - Applying nent
0 0, 0 0 with y(0, 0)
where
is
fi, f 2,
~
0. Since j1 X
gi, g2,
hi, h2,
(0) == y~y 2014 ~x
y and
we
we can
S~
are
vi ) S, is a germ of a vector field with oo flat along [R2 x { 0 }. Let
me
explain
and y ; property
this
a
may
little bit. First
we
assume
that the z-compo-
in the ffollowing 11 win form : write X m
Cx germs and
zero 2014 -component and which az collected all terms linear in x that the z-axis is formally
ii ) together with vi ) reflects
Annales de l’Institut Henri Poincaré -
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159
VECTOR FIELDS ON
invariant; property iii) simply means that hl and h2 don’t contain terms linear in x and y; property vi) indicates that we « absorbed » the ~ terms of the component of X in y. cz Allthough it is not crucial, we can spare some ink and calculations if we observe that we may assume 0. Because if not, we replace X by .
-
this germ is Ceo equivalent with X put the matrix
by means of the identity
map. We want
to
r~/-B
i i
handy form by means of a coordinate change. fl + g2 denote the trace of this matrix. This important role in the sequel. in
a more
Let T
If
we
=
play
an
put r
then
trace will
one
n 1
1
easily checks that
Note, by the way, the characteristic
that 1 (f1 - g2)2 4
equation
+ g1
is 1 4
times the discriminant of
of M. All this suggests the
following coordinate
change : /
As
Vol. 3, n° 2-1986.
1
~
’
160
P. BONCKAERT
straight forward calculation that the new vector in a point (x, y’, z) = ~(x, y, z) can be written in the following from : obtain from
we
a
where hl, h2, y, S ~ satisfy analogous properties as ii), For brevity we write
we can
write down
iii), iv) and vi) above.
~
~
Then
~,~X
in the form
which will become clear in
(new h2) :
perform
For
reasons
z
u~ of the z-axis, just like in the proof of proposition (2. 3. 1)
=
by
means
we always restrict our Calculating straightforward we find
us
a
moment we
a
rescaling case
III,
of the map
Remember that
Let
field
attention to the upper halfspace. if and only if that R*X’ =
simplify the notations by writing again X, y, z,
h2, Q, y, S x instead
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS 0~
of respectively X’,y’, u, h1o R,
put v(u)
=
T(u2~.
Then _
we
h2 R, 2Q - 1, o
something
obtain ,
161
!R 3
1 2 o R, S x
R ; let
a
us
also
of the form: ,
~.
_
where, for the sake of clearness, the following properties hold : /1""’Io.’B.
..... ,.. ~ -...
’"
..
v) Q > 2 a and which is vi) S~ is a vector field with zero --component az along ~2 x {0 }. Let me explain why it is no restriction to assume that
oo
flat
perform a blowing up idea similar to the one in the proof 01ofprothe position (1.1) as follows. Blowing up X n times gives a vector field
We
can
form r /i
1 ~
Sx satisfying similar properties as ii ), iii ) and ri) above. 0, The fact that y(0, 0, 0) # 0 should explain our assumption exist there ..., Q 1} which we take for granted from now on. So
for
h2,
some
and
such that
Vol. 3, n° 2-1986.
162
P. BONCKAERT
~
The to
«
«
strongest »
term in the
~
~
~
expression of X
is
of
y- . We want
course
C/~L weaken » it with respect to the other entries in the matrix
This situation is comparable with the problem : diminish the entries « 1 » emerging in the Jordan normal form of a square matrix. Inspiring ourselves on this, we consider for each P~N the coordinate change
P will be chosen in
is
a
we
For
for
sort
of partial
a
according
moment,
to the
occuring
cases.
In fact ap
blowing up.
have
our
vector
some
field here this
gives :
satisfying property u~)
above.
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
!R~
163
Intermezzo.
Let
me
indicate here
why it was necessary to rescale the z-axis. Take the
example : rr.l
Since the only allowed choices for P are here P 0 and P 1, we cannot weaken the entry « 1 » without creating a similar problem. However after the rescaling we have =
rA
and
by taking P
=
1
we are
End of the intermezzo. We distinguish the cases Case jP1g(0)
Choose P
Here
Since
we
=
lead to the matrix
=
Pi. We distinguish Pi
Q have
=
Vol. 3, n° 2-1986.
11
0.
0 and
0.
=
Subcase Pi
=
0
Q -
1 and
Pi = Q -
1.
(write again x, y, z instead of x, y’, z)
we
may write that
1.
164
P. BONCKAERT
by zP. We can write :
We devide the vector field 03B1P*X 2014 /,
_
2014t ~
B
1
In order to get rid of the possible unboundedness of the terms P hl(x, yzP, z) z 1 and 2P h2(x, yzP, z) it suffices to blow up the latter vector field 2P times z
(without dividing of course); indeed: thanks
to the
property iii) above
we
have ,
if we blow up 2P
P...
,
...11")...
í"B./II
times, the formulas in (II . 3 . 2. 3) imply that
factor z2P. This blowing up construction does not alter the This 1-jet is :
there appears
a
B
i,
,
1
1-jet
in 0
of ~
~
.
Since a 1 # 0, we clearly have a vector field satisfying the assumptions of proposition (2.1.3). The conclusions of proposition (2.1.3) remain valid for our original vector field because of the following reasons :
a) 03B1-1P 1( { (h(z), z) z E ]0, of a
b)
map
oo
if we have
tangent
a cone
oo
[ } ) u { (0,0,0) }
to the
z-axis in 0,
as
is the germ of the well as
graph
K of the form
then 03B1-1P 1 transforms it into r
I
This set contains the __
.
,
.
cone _
_1
Annales de I’Institut Henri Poincaré -
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!R 3
VECTOR FIELDS ON
rn wnicn we nave
or
165
ui the mam theorem
the case; finally the rescaling R transforms this last
accoramg
to
f/.~ .. -B2014m)2 ..
Subcase With
field
P1 = Q a same
rn - -
cone
rI
7
into
.
7
IB. B. ’")
/
D ,
1.
construction
z ( ~ Y~ )
as
in the subcase above
with this time
we
2014component p oz
a
obtain
a
After
’Y( ~ Y
up 2P times (in order to get rid of unbounded terms) field satisfying the assumptions of proposition (1.1). We can conclude just like in the subcase above.
blowing
vector
we
obtain
a vector
Case jP1
g(o) ~
There exist
0.
P2 E ~ 1,
and ~2 ~ 0 such that
...,
_i _2~
Choose P
=
-
~2Po .
I
~/_2Po+2B
P2 . We distinguish three subcases : P2
Pi1
Q - 1,
or
Subcase P2
P1 ~ Q -
1.
We have B
w e aeviae inis vector neia
Vol.
3,
we can
n° 2-1986.
oy
i ~
1
z- ana obtain :
c
1
Agam
.
get
nd ot the unboundedness ot some terms
by blowing up
166
2P
P. BONCKAERT
times, just like above.
Y. The
Denote the result of this
blowing up construction
by
1-jet
of Y is
’"’
-,
0, then the eigenvalues
If a2
I and 0. Hence
are
apply proposition (2 . 2 .1) to obtain the situations II. A or II. B 0» of the main theorem. The same remarks a) and b) in « case jplg(O) hold. If a2 > 0, then the eigenvalues are 0. So up to a linear Y satisfies change of coordinates preserving ~2 x { 0 } and {0}2 the assumptions of proposition (2 .1.1 ). Blowing down this situation, we still have a Cx cone K of finite contact and a 2-dimensional C° subcone S like in situation II. C of the main theorem. Next contains 4 C x 1 cone K’ of finite contact just like in the foregoing case; observe also that 03B1-1P ag preserves the property «having 00 contact with the z-axis » ; S’ is a C° 2-dimensional subcone of K’ ; the rescaling R also preserves the property « having oc contact with the z-axis in 0 » ; R(K’) contains a C’~ cone of finite contact and R(S’) is a C° 2-dimensional subcone of R(K’). All this shows that, for our original vector field, we are in situation II. C of the main theorem.
we can
=
a2 , - ~/~2,
:=
Subcase
P2
We have
Dividing
=
Pi
Q ~
1.
,
this vector field
,
,
by zP r
i
we
obtain : ,,--,
-
Annales de l’Institut Henri Poincaré -
Analyse
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VECTOR FIELDS
mow tms
vector
up
zr times to
field has
a
get
ON R3
167
na 01 ine unoounaea terms,
i ne
resuitmg
1-jet /1
/
v-_,
1
B
~
Since the trace of the matrix
is
a1(~ u)
according Next
we nave ai least one nonzero
we can
Subcase
P2
With
eigenvalue.
rience we can
to the case,
=
propositions (2 .1.1 ), (2 .1. 3), (2 . 2 .1 ) conclude just like in the previous subcase.
(2 . 3 .1 ).
vector field
1z (Xp*X
P1 = Q - 1.
a same
construction
as
above
we
obtain
a
~ with So
a
apply,
or
P
2014- component zy(x, yzP, z).
we can
apply proposition (1.1).
§ 3 . Proof
D
of the main theorem (1.2.1) with of the C° result.
exception
This is merely a summary of all the foregoing. Take a maximal directed sequence of blowing ups of X along D (definition III.1.3). If this sequence is finite, then apply proposition (III.1.4) to obtain situation I. If this sequence is infinite, then, using proposition (III.1.5), D is formally invariant under X. Theorem (111.2.1) implies that, after a finite number of blowing ups, we are led to a vector field (germ) of flatness zero. Now the theorem follows from propositions (1.1), (2.1.1), (2 .1. 3), (2 . 2 .1 ), (2 . 3 .1 ) and (2 . 4 .1 ), since these contain all the possible cases of flatness zero and since we assume that X is non-flat along D. D V. PROOF OF THE C° RESULT IN THE MAIN THEOREM (1.2.1)
§ 1. Definitions
and notations.
We want to provide « universal models » for the situations II. A, II. B and I I . C obtained in the main theorem (1.2.1). For that purpose we introduce : Vol. 3. n° 2-1986.
168
P. BONCKAERT
( 1.1 )
DEFINITION.
-
The
following
germs of vector fields
called the standard models for situation II. A resp. II. B resp. II. C of the main theorem (I . 2 .1 ). are
( 1. 2)
DEFINITION.
-
is called the standard
The germ of the set
cone
around the z-axis.
§ 2. Let us state here in a more the main theorem (I.2.1).
The C° result.
precise
way the assertation announced in
Let X E G3, let D be a direction such that X (21) PROPOSITION. leaves D formally invariant and such that X is non-flat along D. Then there exists a cone K of finite contact around D, a C° cone K’ K~ containing K such that the germ XK’ is C° equivalent with either or SBKS or Sc Ks, according to the fact that X is in situation II . A resp. II. B resp. II. C of the main theorem (1 . 2 . I). Moreover in situation II. B we have C° conjugacy. -
Proof this).
We may
assume
that D is the z-axis
(see part
III for comments
on
know that, after a finite number of blowing ups, after possibly rescaling of the z-axis, after possibly a partial blowing up and after putting in normal form, we are always lead to a vector field Y of one of the following types : (change the sign of X if necessary; with « flat terms » 0 plane) we mean terms Jo flat along the z From part IV
we
a
=
1) the eigenvalue of Y along the z-axis is to
the
z
=
0
plane
is
a
0 and the restriction of Y
hyperbolic expansion; Annales de l’Institut Henri Poincaré -
Analyse
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169
VECTOR FIELDS
with
a>0, b>0, /i(0,0,0) ==~(0,0,0) 0, y(0,0,0)
0;
=
3) the restriction of Y to the z and the z-component of Y is 4) the restriction of Y to the z and the z-component of Y is
=
f(O,O)
=
6)
Y has the
7))
Y
0, g(0, 0, 0) same
3~
form
y, z)y 2014
10)
Y
=
same
y,
form
0;
in 5 but with this time
.f ( ~ Y~
+~
/(0,0,0)>0~(0,0,0)0; 8) Y has the same form 7(0,0,0)0; 9) Y has the 1’(0, 0, 0) 0 ;
0, y(0, 0, 0)
>
as
0
~B
/ ~
with
plane is a hyperbolic expansion z) with y(0, 0, 0) 0, Q > 2 ; 0 plane is a hyperbolic contraction z) with y(0, 0, 0) 0, Q > 2 ;
=
+
y( ~
g(0, 0, 0)
y, z) 2014 + flat +
0;
terms with a >
as
in 7 with this time a
0, f (0, 0, 0)
as
in 7 with this time a
0, f(O, 0, 0)
z)x ~ ~x
+
+
y,z) ~ ~z
0, 0,
>
0,
+ flat terms with
0, a > 0, y(0, 0, 0) 0 (nota bene : we have interchanged the and y compared with the situation in proposition (IV. 2 . 3 .1 )). One has the following table :
j(O, 0, 0) role of
x
Note that the operations (blowing up, rescaling, etc.) are homeomorphisms of [R2 x ]0, oc [ with the property that a sequence tending to the = - 0 plane is transformed into a sequence tending to the z 0 plane. =
Vol. 3, n° 2-1986.
170
P. BONCKAERT
Roughly spoken, the idea is to construct in the origin a local C° equivalence with the corresponding standard model blown up once. Moreover we take care that the homeomorphism, realizing the C° equivalence, transforms sequences tending to the z 0 plane into sequences tending to the z 0 plane. When returning back the whole way this will assure the continuity in (0, 0, 0) of the desired C° equivalence for our original vector field X. We assume that for each considered situation there has been blown up at least once. =
=
Situation II . A. We blow up
This
blowing
once
the standard model SA and get
up transforms the standard
cylinder { (x, y, z) x2
+
y2 - 1, z
>
cone
KsB{ 0 }
into the
(full)
0 }.
out just like in lemmas (IV. 2. 3. 8), (IV. 2 . 3 .10) we can that the z-axis is invariant under Y. We can find a small cylinder B(0, ~c) x ]0, 5 ] around the z-axis such that { (x, y)I X2 + y2 - ,u2 ~ x ]0, 5]] is transversal to the orbits of Y: for types 1 and 3 this is clear from the hyperbolicity (provided decent coordinates are chosen) ; for types 5 and 7 we’ consider the function G(x, y, z) = x2 + y2 and observe that for type 5:
- By flattening
assume
:=
y, z) + flat terms and for type 7 : ( VG(x, y, z), Y(x, y, z) ~ ==2(x~ + 2ax2 + + flat terms ; VG(x, y, z), Y(x, y, z) > == f(.r, ~.7, in both cases we have VG(x, y, z), Y(x, y, z) > > 0 on V/l-,ð provided ~ and 6 are small; hence the orbits of Y are transversal to the level surfaces of G in
Moreover, if 6 is the planes z zo, 0 between Y |V ,03B4 and chosen such that =
To
see
this
we
small, then the orbits in
are also transversal to 03B4. Now it is easy to construct a C° equivalence SA |{x2+y2~ 1 and 0z~03B4}; the homeomorphism h can be
proceed
as
follows. Take the
homeomorphism
Annales de l’Institut Henri Poincaré -
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VECTOR FIELDS ON
Extend n to
171
as follows: u
then its
positive
orbit for Y intersects {(x,y, z) |x2 + y2 /12, 0 z 5} a first time in a point, say, define y 1, yo, zo) to be the intersection of the of orbit the plane z for with negative SA z1) Zo; finally define for 0 ð. 0, 20) = (0, 0, zo) zo =
N
=
Situation 77. B. We blow up the standard model
SB and get ~
N
1 nis vector neia is transversal to tne 1-1 _ 7 ~v
1x .
,1
.,
I v-2
-
nanspnere.
-v-2 _J ,,2 ,I
,,2
....L
transversal 10 me orons 01 Y : ior
....L
type
-2 _ A
no...
’7~ 2014 ~~ ~~, 4 tms ciear from the
hyperbolicity
and from y(0,0,0) 0 (provided decent coordinates are types 6 and 8 we consider the function G(x, y, z) = x2 + observe that for type 6:
m both
~BB
..
~,
_
1_
~t100
/~~ /
Consider the
homeomorphism .
--
--
I
y2
’)....0+ 1...~~....
+
z2 and
_B
U on a small set
cases ( VCJ(x, y, z), Y(x, y, z) > ~~
,.,.B
w
chosen) ; for
.. .
~~
...
2
, "’B
..
Extend h to Vð as follows. Let and 03C6B denote the flows of Y resp. SB’ For each (x, y, z) E Va there exists a t >_ 0 such that ~y(2014~ (x, y, z)) ~ Hi. We force the conjugacy to be true on Va by defining LI - -
we cnecK me
-
-B B
JL
required property
/.
1_l 1 /~l__ __ _B111 B
ior
yi,
oe a
sequence
m
v03B4
denote the sequence 0). Let (ti times » such that ~(2014f~ Suppose by contradiction y~ the z-components of a subsequence would stay
tending to the z of « that
-
Vol. 3, n° 2-1986.
=
0 plane (that is :
lim zl
=
172
I
P. BONCKAERT
M
>
0 for
some
constant M. This
implies
that sup tik
+
Hence
~6~
ie
z-components of ~y(2014~ (x~k, ylk,
o we
have
a
decent
conjugacy
h between
tend to zero. A contradiction. and
preimage h 1( ~ (x, y, z) + y2 1 and z > 0}) still contains a ylinder around the z-axis in Va. Returning this situation the whole way ~ack to our original vector field X this gives us the desired cones and onjugacy. .’he
ituation II . C. We blow up the standard model
Sc and get
1 the level surfaces of g are transversal to the Hence in the region z of orbits Sc. Take 0 b~ 1. Take a2 > 0 so small that the intersections of the level surface G-1( - ~2) with the planes y a2 and y a2 are circles C2 z the 1. See inside 2 from on. now figure region resp. D2 =
FIG. 2.
-
=
-
Sketch of the first octant. Annales de l’Institut Henri Poincaré -
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173
VECTOR FIELDS
We also consider the level surface G ~(1); this is a two-leaved hyperboloid. Next we consider the surface obtained by letting flow all the points of the circle C2 until they hit the level surface G - l( 1 ). We do the same x ]0, x [ a region R2. thing for the circle D2. In this way we enclosed in we that the out assume 0 By flattening may y plane is invariant under Y. We want to make an analogous construction as above for Y. We have for type 2: =
and for type 9:
ana tor
type
1U:
/ K7/U/,,
..
_B
So for all three types we can find in V:= W n ([R2 x ]0, t [ ) :
a
neighbourhood W
of (0,
0, 0)
such that
miniature version of the construction for Sc. 0 we can make the following construction inside V. Consider the level surface G - 1( ð 1)’ The intersections of it with the planes
we
want to
make
For small
5i
a
>
-
~ v
=
5i
and y = -03B42 03B12
surface obtained
by letting
hit the level surface
are
circles
flow all the
G-1(03B41 03B42).
Ci
resp.
points
We do the
Di. Consider also the
of the circle C i until
same
r
Second
we
Vol. 3. n° 2-1986.
extend h to
for the circle Di.
thing
In this way we enclosed a region Ri in V. Now we try to make an equivalence between define the homeomorphism
and ~
ç:
Ras follows. Require that
a
they
Sc
First
we
Î
level surface 7
174
P. BONCKAERT
(a1 e 2014 5i, 2014 )
is
mapped
onto the level
(x, y, z) E Ri lying in a level surface where the negative integral curve of time the level surface
of
G-1 (03B42 03B41 ai
we
surface
,
for
y 1, Z1)
consider the point
through (x, y, z) hits for the first we define G"~(2014 5i); h(x, y, z) to be the intersection
with the
Y
yB zl .
positive integral curve of Sc through
We check the required property for h. 0. sequence in Ri with lim zi
Suppose
that
(xi,
is
yi,
a
=
Let (xI
, y1i, z1i) be the point where the negative integral curve of Y through
hits for the first time the level surface G’~(2014 tends to the y 0 plane, we get that sequence (xl, yl, 0 plane. tends to the z D
~i).
Since the
=
yi,
=
(2 . 2) [Cam ] .
REMARK. - We have used
VI. SOME
some
ideas
comparable
to
those in
EXAMPLES, COUNTEREXAMPLES QUESTIONS
AND SOME
§
1. A
counterexample
and
some
questions.
might pose the question whether every germ in 0 E [R3 of a Cx satisfying a ojasiewicz inequality (see definition I.1.17) one-dimensional invariant manifold, or equivalently, possesses a whether it possesses an integral curve tending to 0 (in positive or negative time) in a Ceo way (by tending to 0 in a Cr way we mean : if we add the origin to the integral curve, we obtain a C~ invariant manifold). The answer is no. We give an example of a germ for which no integral curve can tend to zero in a C2 way; the germ has nonzero 1-jet. One
vector field
(1.1)) fies
EXAMPLE.
-
Let X
=
+
+
ojasiewicz inequality. No orbit of X Proof - a) Blow up X in the x-direction: a
x2 ~ ~z. can
Observe that X satis-
tend to 0 in
a
C2 way.
Observe that
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VECTOR FIELDS ON
has
just
one
singularity
in
(0, 0).
!4°
175
Blow up X in the x-direction : ,
-,
,
Observe that
nas no
singularities. up X in the
h) Blow
y-direction : ~
Observe that
has 11U
sing
c) Blow
up X in the z-direction : ,
Observe that
~~~-~
Now
we are
by blowing
able to describe the vector field X on S2 spherically (see for example II. § 1 or [Ta,
up X
obtained Du2] for defi-
nition and construction). The only singularities of X IS2 x {0} are (1, 0, 0) and ( - 1,0,0). If an orbit 811 of X tends to 0 in a C2 way, then X must have an orbit 82 which tends in 1 a C~ way to ( 1, 0, 0) or ( -1, 0, 0) since these are the only singularities on S2 x { 0 }. If we blow up X in (1, 0, 0) or ( - 1,0,0) we don’t have a singularities any more, except in the « corner » : see figure 3. This contradicts the fact that 82 tends C1 to (1, 0, 0) or ( -1, 0, 0). Q
( 1. 2) REMARK. - The vector field X in example ( 1.1 ) must have an orbit in the first octant(.v, y, ~) E [R3I x >_ 0, y >- 0, z > 0 } tending to 0 in a C° way. One can see this as follows. Denote Vl = {(x, y, z) E [R3 (x 0 or y 0 or z 0) and x > 0 and and and and z > 0 r > 0 x+y+z 1} V2 == { (x, y, z) e ~ ! jc > 0 =
and y > 0 and Vol. 3, n° 2-1986.
z >
=
=
0 }. V3 is the first octant. See figure 4.
176
P. BONCKAERT
FIG. 3.
-
Blow up X in
(1,0, 0).
FIG. 4.
Each t
-
point of vl B ~ 0} +00. This follows
the function
G(x,
y,
enters the first octant
from the Since
immediately
z) x + y + z. VG(,r, y, z), X(x, y, z) ) = =
V3 and
y +
z
leaves it for of X. Consider
never
expression + x2
Annales de l’Institut Henri Poincaré -
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177
VECTOR FIELDS
the
positive
consider the
orbit of
a
point
.-........-
""""’4.""""’’’’’’------
T(u) _ -
V1 B { 0 } intersects V2 L,)f 0 } of a point
negative orbit {
£
If
of
‘,.~, .. ~ J
’JJ, then
T/,.B _
.
lim
necessarily
r) leaves V3 in negative time in
some
’B1 in t ~
a
point
or
V2
v E
to 0 if
to the
lim
u)
-
r E
-.---......
point,
---"’’’’’’’’’0
0. If
=
Conversely,
V~. Put
..~- 1 - n
I
-v-
L --, ..&. s...1..~,..__J
once.
----
T(v)
---------------
> - ex)
~_.
then
say,
r.- 1,; ,
point of Vi where the negative orbit through v hits Vi v) 0. Certainly ViB { 0 } since the positive =
orbit of each point of V1B{ 0 } hits V2. Since V2 is compact and since V1B{ 0 } is not compact, necessarily h(V2) V1. Hence there must be a point in V2 tending to 0 for t - - 00. =
I don’t know whether this vector field X possesses (1 . 3) QUESTIONS. orbit tending in a C~1 way to 0. I generalize this question : does there exist a germ in 0 E [R3 of a Cx vector field satisfying a xojasiewicz inequality without an orbit tending I to 0 in a C° way? in a C~ way? (in positive or negative time). -
an
vector fields having a Coo dimensional invariant manifold.
§ 2 . Examples of one
A very
general observation from
the
chapters
III and IV is :
OF THE PROOF OF THE MAIN THEOREM. If X E G3 D direction leaves formally invariant (that is: invariant by the oc jet) and if X is non-flat along D then there exists a C x one-dimensional invariant manifold x tangent to D. D If we consider germs in 0 E [R3 of vector fields satisfying a Mojasiewicz inequality, it hence suffices to look for conditions which guarentee the existence of a formally invariant direction. In general it may be a difficult task to investigate whether a given vector field has a formally invariant direction. Some tools for this can be : apply the normal form theorem (which we will do in some examples hereafter); or: try to blow up the vector field until you find a singularity having a formally invariant direction by the normal form theorem. An important result in this sense is :
(2.1) CONSEQUENCE a
(2.2) THEOREM [B. D. ]. - If X E G3 satisfies a ojasiewicz inequality DX(0) has eigenvalues 0, i i~. - i i ( i E (1~ B ~ 0 } ) then X has a C x
and if A
:=
3. n ~-1 y~b.
178
P. BONCKAERT
one-dimensional invariant manifold D tangent in 0 to the rotation axis of etA, t # 0. Moreover there exists a cone of finite contact around D in which we have situation II. A or II. B of the main theorem (1.2.1). In the same spirit we can apply the normal form theorem to other types of singularities with nonzero 1-jet. For many cases the result is well known, but let us list them for the sake of completeness :
(2 . 3)
MORE
EXAMPLES.
-
Let X e G, Xi = DX(0).
Annales de l’lnstitut Henri Poincaré -
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Proof.
179
2014 Denote _
where
I~$3
A,
are
_
(~
.
real constants r~
_
~
(zero rv
is
1
allowed).
Put
i
where h > 2 (see formulation of normal form theorem also denote, for h >_ 2, 0 j h, 0 i _ j :
(I. 1 21 )).
Let
us
i ms is a oasis ior n .
After
a
trivial calculation
one T I
finds that B
where
So D has On Hh
a
diagonal matrix with respect to this basis. put the standard inner product with respect
we
Vol. 3. n° 2-1986.
to
this basis.
180
P. BONCKAERT
For as
O? c002 ~ 0. example 1 we see that complementary space (Im D)1; an element
2014
contain the terms e001 = zh
nor
3.B’
e002 = zh
e002 ~ Im D. Take
so
(Im D)1
of
2014
cannot
since
~y
Hence in the normal form the z-axis is formally invariant. For example 2 the same reasoning applies; for example 5 we see that ehn3 E Im D and we can follow an analoguous reasoning. A similar method works for examples 3, 4, 6. Next we calculate that
so
S 1 has a matrix of the form
where A is an upper triangle matrix with zero diagonal elements is nilpotent). Then one easily calculates that S 1 must be nilpotent. So for examples 7, 8, 9 we can write
(so A
where D is semi-simple and Si is nilpotent. The fact that Im D c Im [X 1 - ]~ implies, in the same way as above, the results claimed in examples 7, 8 and 9. Concerning example 10 we find that the matrix of S2 is of the form
where B is an upper has a matrix
triangle matrix r
and
one
easily
..
-
with vi
calculates that S3 must be
zero
diagonal
elements. So 83
/~ ~)
nilpotent.
Annales de l’Institut Henri Poincaré -
Analyse
non
linéaire
181
VECTOR FIELDS
[X 1, - ]~ D + S3 again reasoning as above can be made. For examples 11 and 12 we find As
=
,
so
e001, e002~ Im
[Xi, 2014 ]h
1m D
c
1m
[7(1, - Jh
and
the
same
that ’B
~
and
we can reason
like before.
D
In case j1X(0) 0 there is, as far as I know, very (? . 4) REMARK. little known on normal forms. But as already said, blowing up may sometimes help. This is the case for a C x gradient vector field X = grad f : F. Takens 0 then X has showed me, using a blowing up argument, that a ex invariant manifold, in all (finite) dimensions. In example (1.1) 0 was not an isolated zero of the first nonvanishing jet. Even the assumption that 0 is an isolated zero of the first nonvanishing jet (which implies that the radial eigenvalue in a singularity of X IS2 x {0} is nonzero) is not enough to have an invariant direction : =
-
EXAMPLE.
(2.5)
Observe that 0 is
-
an
Let
isolated
zero
of J2X(0). No orbit of X
can
tend to 0
in a C2 way
Proof - a)
Blow up X in the x-direction : a
i
i
nas
just
one singularity
m
(U, UJ.
Blow up Xx in the x-direction :
and observe that
has
no
singularities.
3. n° 2-1986.
~
,
182
?. BONCKAERT
b) Blow up X in the y-direction : and observe that
has
no
c)
singularities.
Blow up X in the z-direction : ~
nd observe that
has no singularities. Now we can conclude
just
like in
(2.6) .REMARK. This example This can be seen from the expression
example (1.1).
D
to 0 in a C~ way. is a hyperbolic expansion.
has orbits tending
(2. 7) SOME FINAL REMARKS. - a) Concerning the main theorem (1.2.1): when X is analytic, it is not necessary that (one of) the obtained invariant manifold(s) is also analytic, as was pointed out to me by the referee: for X
=
x - + ((yY - z2 a ~+ z"2014OZ
the z-axis have
an oo
all the invariant directions tangent to g
jet in 0 of the form
b) Concerning proposition (V . 2 .1) about C° equivalence with standard (but cannot prove) that « C° equivalence » can be « C° replaced by conjugacy ». models: I presume
REFERENCES [Ar] [A. M.] [A. R.] [B. D.]
[Be]
V. ARNOLD, Équations différentielles ordinaires, Éditions Mir, Moscou, 1974. R. ABRAHAM and J. E. MARSDEN, Foundations of Mechanics. Benjamin, London, R. ABRAHAM and J. ROBBIN, Transversal Mappings and Flows. Appendix by Kelley, Benjamin, New York, 1967. P. BONCKAERT and F. DUMORTIER, Smooth Invariant Curves for Germs of Vector Fields in R3 whose Linear Part Generates a Rotation, to appear in
Journal of Differential Equations. G. R. BELITSKII, Equivalence and Normal Forms of Germs of Smooth pings. Russian Math. Surveys, t. 33, 1978, p. 107-177. Annales de l’Institut Henri Poincaré -
Analyse
non
Maplinéaire
VECTOR FIELDS ON
~3
183
H. W. BROER, Formal Normal Form Theorems for Vector Fields and some Consequences in the Volume Preserving Case, in: Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Math., t. 898. 1981. [Cam] M. I. T. CAMACHO, A contribution to the Topological Classification of Homogeneous Vector Fields in R3 (preprint). [Car] J. CARR, Applications of Centre Manifold Theory, Springer Verlag, New York, 1981. [Di] J. DIEUDONNÉ, Foundations of Modern Analysis, Academic Press, New York, 1969. [Du1 ] F. DUMORTIER, Singularities of Vector Fields on the Plane. Journal of Diff. Eq., t. 23, 1977, p. 53-106. [Du2] F. DUMORTIER, Singularities of Vector Fields. Monografias de Matematica, t. 32, IMPA, Rio de Janeiro, 1978. [D. R. R.] F. DUMORTIER, P. R. RODRIGUES and R. ROUSSARIE, Germs of Diffemorphisms in the Plane. Lecture Notes in Math., t. 902, 1981. [Go] R. E. GOMORY, Trajectories tending to a Critical Point in 3-space. Annals of Math., t. 61, 1955, p. 140-153. [Gu] J. GUCKENHEIMER and P. HOLMES, Nonlinear oscillations, Dynamical Systems and Bifurcations, Springer-Verlag, New York, 1983. [Ha] P. HARTMAN, Ordinary Differential Equations. Wiley and Sons, New York. [H. P. S.] M. W. HIRSCH, C. C. PUGH and M. SHUB, Invariant Manifolds. Lecture Notes in Math., t. 583, 1977. [Ir] M. C. IRWIN, Smooth Dynamical Systems, Academic Press, New York, 1980. [Ke] A. KELLEY, The Stable, Center-Stable, Center, Center-Unstable, Unstable Manifolds. Journal of Diff. Eq., t. 3, 1967, p. 546-570. [M. M.]J. E. MARSDEN and M. McCRACKEN, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [Na] N. NARASIMHAM, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam. 1968. [N. S.] V. V. NEMYTSKH and V. V. STEPANOV, Qualitative theory of Differential Equations. Princeton, Princeton University Press, 1960. [P. M.] J. PALIS and W. DE MELO, Geometric Theory of Dynamical Systems, SpringerVerlag, New York, 1982. [Sh] D. S. SHAFER, Singularities of Gradient Vectorfields in R3. Journal of Diff. Eq., t. 47, 1983, p. 317-326. [Sm] D. SMART, Fixed point theorems. Cambridge University Press, 1974. [Ta] F. TAKENS, Singularities of Vector Fields. Publ. Math. IHES, t. 43, 1974, p. 47-100. [We] J. C. WELLS, Invariant Manifolds of Non-Linear Operators. Pacific Journal of Math., t. 62, 1976, p. 285-293. [Ya] S. YAMAMURO, Differential calculus in topological linear spaces. Lecture Notes in Math., t. 374, 1974.
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